Harmonic measures versus quasiconformal measures for hyperbolic groups

HARMONIC MEASURES VERSUS QUASICONF ORMAL MEASURES F OR HYPER BOLIC GR OUPS S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINSKY & PIERRE MA THIEU Abstract. W e establish a dimensio n formula for the harmonic meas ure of a ﬁnitely s up- po rted and symmetric r andom walk on a hyper bo lic group. W e a lso characterize random walks for whic h this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric po int o f view on random walks a nd, in pa rticular, which allows us to interpret ha rmonic measures as quasiconformal measures on the b ounda r y of the group. 1. Introd uction It is a leading thread in h yp erb olic geometry to try to understand prop erties of h yperb olic spaces b y studying their larg e- scale b ehaviour. This principle is applied through the in tro- duction of a canonical compactiﬁcation whic h c haracterises the space itself. F or instance a h yperb olic group Γ in the sense of Gro mov admits a natural b oundary at inﬁnity ∂ Γ: it is a topolo gically well-deﬁne d compact set on whic h Γ acts b y homeomorphisms . T ogether, the pair consisting of the boundary ∂ Γ with the action of Γ ch aracterises the hyperb o licit y o f the group. T op ological prop erties of ∂ Γ a lso enco de the algebraic structure of the group. F or instance one prov es that Γ is virtually free if and only if ∂ Γ is a Can tor set (see [46 ] and also [12] for other results in this v ein). Moreo v er, the b oundary is endo w ed with a canonical quasiconformal structure whic h determines the quasi-isometry class of the group ( see [31] and the references therein for details). Characterising sp ecial sub classes of hy p erb olic groups suc h as co compact Kleinian gro ups often req uires the construction of sp ecial metrics and measures on the b oundary whic h carry some geometrical information. F or example , M. Bonk and B. Kleiner pro v ed that a group admits a co compact Kleinian action on the hy p erb olic space H n , n ≥ 3, if and only if its b oundary has topo lo gical dimension n − 1 and carries an Ahlfors-regular metric of dimens ion n − 1 [9]. There are tw o main constructions of measures on the b oundary of a h yp erb olic g r oup: quasiconformal measures and harmonic measures. Let us recall these constructions. Giv en a co compact prop erly discon tinuous action of Γ by isometries on a p oin ted prop er geo desic metric space ( X , w , d ), the Patterson-Sulliv a n pro cedure consists in taking w eak limits of 1 P γ ∈ Γ e − sd ( w ,γ ( w )) X γ ∈ Γ e − sd ( w ,γ ( w )) δ γ ( w ) Date : May 2 8, 2018. 2000 Mathe matics Subje ct Classiﬁc ation. 20 F6 7, 60B 15 (11 K 55, 20F69 , 28A75 , 60J5 0, 60J 65). Key wor ds and phr ases. Hyp er bo lic groups, random walks o n gro ups, har monic mea sures, quasiconformal measures, dimens io n of a mea s ure, Martin b o undary , Brownian motion, Gr een metr ic . 1 2 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU as s decreas es to the logarithmic volume grow th v def. = lim sup R →∞ 1 R log | B ( w , R ) ∩ Γ( w ) | . P atterson-Sulliv an measures are quasiconformal measures and Hausdorﬀ measures of ∂ X when endo w ed with a visual metric. Giv en a probabilit y measure µ on Γ, the random w alk ( Z n ) n starting from the neutral elemen t e associated with µ is deﬁned b y Z 0 = e ; Z n +1 = Z n · X n +1 , where ( X n ) is a seque nce o f indep enden t and iden tically distributed r a ndom v ariables of la w µ . Under some mild assumptions on µ , the w alk ( Z n ) n almost surely con v erges to a p oint Z ∞ ∈ ∂ Γ. The la w of Z ∞ is by deﬁnition the harmonic meas ure ν . The purp ose of this w ork is to inv estigate the interpla y b etw een those tw o classes of mea- sures and tak e adv antage of this in terpla y to deriv e information on the geometry of harmonic measures. The usual to ol for this kind o f results is to replace the action of the gro up by a linear-in-time action of a dynamical sys tem and then to apply the thermo dynamic formalism to it: for free groups and F uc hsian groups, a Mark o v-map F Γ has b een intro duced on the b oundary whic h is orbit-equiv alent to Γ [13, 38]. F or discrete subgroups of isometries of a Cartan-Hadama r d manifold, one ma y w ork with the geo desic ﬂow [36, 37, 26, 28]. Both these metho ds seem diﬃcult to implemen t for general hy p erb olic groups. On the one hand, it is not ob vious ho w to asso ciate a Mark o v map with a general h yp erb olic g roup, ev en using the automatic structure of the gr oup. On the other hand, the construction o f the geo desic ﬂo w for general h yperb olic spaces is delicate and its mixing prop erties do not seem strong enough to apply the thermo dynamic formalism. In a diﬀeren t spirit, it is prov ed in [14] that a n y Patters on-Sulliv an measure can b e realized as the harmonic measure of some random walk . Unfortunately suc h a general statemen t without an y information on the la w of the incremen ts of the random w alk is not suﬃcien t to pro vide a n y real insigh t in t he b eha viour of the walk. Our approac h directly com bines geometric and probabilistic arguments. Since we a v oid using t he t hermo dynamic formalism, w e b eliev e it is more elemen tary . W e mak e a hea vy use of the so-called Green metric asso ciated with the random w alk, and w e emphasize the connections b et w een the geometry of this metric a nd the prop erties of the random walk it comes from. The problem of iden tifying the Martin and vis ual b oundaries is an ex ample of suc h a connection. W e g iv e suﬃcien t conditions for the G reen metric to b e h yp erb olic (although it is not geo desic in general). On the other hand, its explicit expres sion in terms of the hitting probabilit y of the random w alk mak es it p ossible to directly ta ke adv an t a ge of the indep endence of the incremen ts of t he w a lk. The com bination of bot h facts yields v ery precise estimates on ho w random paths deviate from geo desics. Th us w e sho w that , for a general hy p erb olic group, the Hausdorﬀ dimension of the harmonic measure can be explicitly computed and satisﬁes a ’dimension-en trop y-rate of escap e’ form ula and w e c haracterise those harmonic measures of maximal dimension. Our p oin t of view also allo ws us to get an alternativ e and rather straightforw ard pro of of the fact that the harmo nic measure of a ra ndo m w alk on a F uchs ian group with cusps is singular, a result previously established in [20] and [16] b y completely diﬀeren t methods. HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 3 The r est of this in tro duction is dev oted to a more detailed desc ription of our results . 1.1. Geometr ic sett ing. G iv en a hyperb olic group Γ, w e let D (Γ) denote the collection of h yperb olic left-in v ariant metrics o n Γ whic h are quasi-isometric to a word metric induced b y a ﬁnite generating set of Γ. In general these metrics do not come from prop er geo desic metric spaces as w e will see (cf. Theorem 1 .1 for instance). In the sequel, w e will distinguish the group as a space and as acting on a space: w e k eep the notat ion Γ for the gro up, and w e denote b y X the gr o up as a metric space endow ed with a metric d ∈ D (Γ). W e may equiv alently write ( X , d ) ∈ D ( Γ ) . W e will often r equire a base p oin t whic h w e will den ote b y w ∈ X . This setting enables us to capture in particular the following tw o situations. • Assume that Γ admits a co compact prop erly discontin uous action b y isometries on a prop er geo desic space ( Y , d ). Pic k w ∈ Y suc h that γ ∈ Γ 7→ γ ( w ) is a bijection, and consider X = Γ( w ) with the restriction of d . • W e may c ho ose ( X , d ) = (Γ , d G ) where d G is the Green metric asso ciated with a random w alk ( see Theorem 1.1). Let µ b e a symmetric probability measure the supp ort of whic h generates Γ. Eve n if the suppo r t of µ ma y b e inﬁnite, w e will require some compatibilit y with the geometry of the quasi-isometry class o f D ( Γ ) . Th us, we will often assume one o f t he following tw o assumptions. Giv en a metric ( X , d ) ∈ D (Γ), w e sa y that the rando m w alk has ﬁnite ﬁrst moment if X γ ∈ Γ d ( w , γ ( w )) µ ( γ ) < ∞ . W e say that the random walk has an exp onen tial moment if t here exists λ > 0 suc h that X γ ∈ Γ e λd ( w,γ ( w )) µ ( γ ) < ∞ . Note that b oth these conditions o nly dep end on the quasi-isometry class of the metric. 1.2. T he Green metr ic. The analo gy b etw een b o th families o f measures – quasiconformal and harmonic – has already b een p o in ted out in the literature. Our ﬁrst task is to make this empirical fact a theorem i.e., w e prov e that harmonic measures a re indeed quasiconformal measures for a w ell-c hosen metric: giv en a s ymmetric law µ on Γ suc h that its s upp ort generates Γ, let F ( x, y ) be the probabilit y that the random w alk started at x ev er hits y . Up to a constan t factor, F ( x, y ) coincides with the Green function G ( x, y ) def. = ∞ X n =0 P x [ Z n = y ] = ∞ X n =0 µ n ( x − 1 y ) , where P x denotes the proba bilit y law of the random w alk ( Z n ) with Z 0 = x (if Z 0 = e , the neutral elemen t of Γ, w e will simply write P e = P ), and where, for eac h n ≥ 1, µ n is the la w of Z n i.e., the n th conv olution p ow er of the measure µ . W e deﬁne the Gr e en metric b et w een x and y in Γ b y d G ( x, y ) def. = − log F ( x, y ) . This metric was ﬁrst in tro duced b y S. Bla c h ` ere and S. Broﬀ erio in [7] and further studied in [8]. It is w ell-deﬁned as so on as the walk is transien t i.e., eve n tually lea v es a n y ﬁnite set. This is the case as soon as Γ is a non-elemen tary h yperb o lic group. 4 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU Non-elemen tary h yp erb olic groups are non-amenable and for suc h groups and ﬁnitely sup- p orted la ws µ , it w as pro v ed in [7] that the Green and word metrics are quasi-isometric. Nev ertheless it do es not follo w from this simple fact that d G is h yp erb olic, see t he disc ussion b elo w, § 1.7. W e ﬁrst pro v e the follo wing: Theorem 1.1. L et Γ b e a n on-elementary hyp erb olic gr oup, µ a symmetric pr ob ability me as ur e on Γ the supp ort of which gener ates Γ . (i) Assume that µ has an exp one n tial momen t, then d G ∈ D (Γ) if and only if for any r ther e exists a c onstant C ( r ) such that (1) F ( x, y ) ≤ C ( r ) F ( x, v ) F ( v , y ) whenever x, y and v ar e p oints in a lo c al ly ﬁnite Cayley gr aph of Γ and v is at distanc e at most r fr om a ge o desi c se gm e nt b etwe en x and y . (ii) If d G ∈ D (Γ) then the harmon i c me asur e is A hlfors r e gular of dimension 1 /ε , when ∂ Γ is endowe d with a v i s ual metric d G ε of p ar am e ter ε > 0 induc e d by d G . Visual metrics are deﬁned in the next section. A. Ancona pro v ed that (1) holds for ﬁnitely supp orted law s µ . Condition (1) has also b een coined b y V. Kaimanovic h as the key ingredien t in proving that the Martin b oundary coincides with the geometric (h yp erb olic) boundary [28, Thm 3.1] (See also § 1.5 and § 3 .2 for a further discussion on the relationships b et w een t he Gr een metric and the Martin boundar y). Theorem 1 .1 in particular yields Corollary 1.2. L et Γ b e a non-elementary hyp erb olic gr oup, µ a ﬁ n itely supp orte d symmetric pr ob ability me asur e on Γ the supp ort of whic h gene r ates Γ . Then its asso ciate d Gr e en m etric d G is a left-invariant hyp erb olic metric on Γ quasi-is o metric to Γ such that the harmonic me asur e is Ahlfors r e gular of dimension 1 /ε , when ∂ Γ is endow e d with a vis ual metric d G ε of p ar ameter ε > 0 induc e d by d G . Our second source of examples of r a ndom w a lks satisfying (1) will come from Brownian motions on Riemannian manifolds of negativ e curv ature. The corresp onding law µ will t hen ha v e inﬁnite support (see § 1.6 a nd § 6) . 1.3. D imension of the harmonic measure at inﬁnit y. Let ( X , d ) ∈ D (Γ). W e ﬁx a base p oin t w ∈ X and consider the random walk on X started at w i.e., the sequence of X -v alued random v ariables ( Z n ( w )) deﬁned by the action of Γ on X . There are (a t least) t w o natural asymptotic quan tities one can consider: the as ymp totic e n tr opy h def. = lim n − P γ ∈ Γ µ n ( γ ) log µ n ( γ ) n = lim n − P x ∈ Γ( w ) P [ Z n ( w ) = x ] lo g P [ Z n ( w ) = x ] n whic h me asures the wa y the la w of Z n ( w ) is spread in diﬀeren t directions, and the r ate o f esc ap e or drif t ℓ def. = lim n d ( w , Z n ( w )) n , whic h estimates how far Z n ( w ) is from its initia l p oint w . (The ab o v e limits for h and ℓ are almost sure and in L 1 and t hey are ﬁnite as so on as the la w has a ﬁnite ﬁrst momen t.) W e obt a in the follo wing. HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 5 Theorem 1.3. L et Γ b e a n on-elementary hyp erb olic g r oup, ( X, d ) ∈ D (Γ) , d ε b e a visual metric of ∂ X , and let B ε ( a, r ) b e the b al l of c enter a ∈ ∂ X and r adius r for the distanc e d ε . L et ν b e the harmonic me asur e of a r andom walk ( Z n ) whose incr ements ar e given by a symmetric law µ with ﬁnite ﬁrst moment such that d G ∈ D (Γ) . The p ointwise Hausdorﬀ dimension lim r → 0 log ν ( B ε ( a,r )) log r exists for ν -almost every a ∈ ∂ X , and is indep e n dent fr o m the choic e of a . Mor e pr e cisely, for ν -almost every a ∈ ∂ X , lim r → 0 log ν ( B ε ( a, r )) log r = ℓ G εℓ wher e ℓ > 0 denotes the r ate of esc a p e of the walk with r esp e ct to d and ℓ G def. = lim n d G ( w, Z n ( w )) n the r ate of esc ap e with r esp e ct to d G . W e recall that the dimension of a measure is the inﬁm um Hausdorﬀ dimension of sets of p ositiv e measu re. In [8], it w as sho wn that ℓ G = h the asymptotic entrop y of the w a lk. F rom Theorem 1 .3, w e deduce that Corollary 1.4. Under the as s ump tions of The or em 1.3, dim ν = h εℓ wher e h denotes the asymptotic entr opy of the walk and ℓ its r ate of esc ap e with r esp e ct to d . This dimension form ula already app ears in the work o f F. Ledrappier for random w alks on free groups [38]. See also V. K a imano vic h, [29]. F or general h yp erb olic g roups, V. Leprince established the inequalit y dim ν ≤ h/ ( εℓ ) and made constructions of harmonic measures with arbitrarily small dimension [34 ]. More recen tly , V. Leprince established that h/ε ℓ is also the b ox dimension of the harmonic measure under t he sole a ssumption t ha t the random w alk has a ﬁnite ﬁrst moment [35]. Not e ho w ev er that the notion of box dimension is to o w eak to ensure the existenc e of the p oin t wise Hausdorﬀ dimension almost ev erywhere. This formula is also closely related to the dimension form ula prov ed for ergo dic inv ariant measures with p ositiv e entrop y in the con text of g eometric dynamical systems: the drift corresp onds to a Lyapuno v exp onen t [50]. 1.4. C haracterisation of harmonic measures with maximal dimension. Given a ran- dom w alk on a ﬁnitely generated group Γ endo w ed with a left-in v arian t metric d , the so-called fundamental ine quality betw een the asymptotic en trop y h , the drift ℓ and the logarithmic gro wth ra te v o f the a ction of Γ reads h ≤ ℓ v . It holds as so on a s all these ob jects are w ell-deﬁned (cf. [8]). Corollary 1.4 provid es a geometric in terpretation of this inequalit y in terms of the harmonic measure: indeed, since v /ε is t he dimension o f ( ∂ X , d ε ), see [15], it is clearly larger than the dimension of ν . A. V ershik suggested the study of the case of equalit y (see [19, 48]). F or an y hy p erb olic group, Theorem 1.1 implies t ha t the equality h = ℓv holds for the G reen metric and Theorem 1.5 b elo w sho ws that t he equality for some d ∈ D (Γ) implies d is almost prop or t io nal to d G . In particular, g iv en a metric in D (Γ), all t he harmonic measures for whic h the (fundamen ta l) equalit y holds b elong to the same class of quas iconformal measures. In the sequel, t w o measures will b e called e quivalen t if they share the same sets o f zero measure. 6 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU Theorem 1.5. L et Γ b e a non-elementary hyp erb olic gr oup and ( X , d ) ∈ D (Γ) ; let d ε b e a visual metric of ∂ X , and ν the harmonic me asur e given by a symmetric l a w µ with an exp onen tial moment, the supp ort of w hich gener ates Γ . We further assume that ( X, d G ) ∈ D (Γ) . We denote by ρ a quasic onformal me asur e on ( ∂ X , d ε ) . The fol low ing pr op ositions ar e e quivalent. (i) We have the e quality h = ℓv . (ii) The me asur es ρ and ν ar e e quivalent. (iii) The me a s ur es ρ and ν ar e e quivalent and the density is almost sur ely b ounde d a n d b ounde d away fr om 0 . (iv) T he map (Γ , d G ) I d − → ( X , v d ) is a (1 , C ) -quasi-isometry. (v) T he me a sur e ν is a quas i c onformal me asur e of ( ∂ X , d ε ) . This theorem is the counterpart of a result of F. Ledrappier for Brownian motions on uni- v ersal cov ers o f compact Riemannian manifolds of negative sectional curv ature [36], see also § 1.6. Similar results hav e b een established for the free group with free g enerators, see [38]. The case of equalit y h = ℓv has a lso been studied for particular sets of generators of free pro ducts of ﬁnite groups [41]. F or univ ersal co vers of ﬁnite graphs, see [39]. Theorem 1.5 enables us to compare random walks and decide when their harmonic measures are equiv alen t. Corollary 1.6. L et Γ b e a no n -elementary hyp erb olic gr oup with two ﬁnitely s upp orte d s ym- metric pr ob ability me asur es µ and b µ wher e b oth supp orts gener ate Γ . We c onsider the r ando m walks ( Z n ) and ( b Z n ) . L et us d e note their Gr e en functions by G and b G r esp e ctively, the asymp- totic en tr opies by h and b h , and the harmonic me asur es se en fr om the neutr al eleme n t e by ν and b ν . Th e fol lowin g p r op ositions ar e e quivalent. (i) We have the e quality b h = lim − 1 n log G ( e, c Z n ) in L 1 and almost sur ely. (ii) We have the e quality h = lim − 1 n log b G ( e, Z n ) in L 1 and almost sur ely. (iii) The me asur es ν and b ν a r e e quivalent. (iv) T her e is a c onstant C such that 1 C ≤ G ( x, y ) b G ( x, y ) ≤ C . 1.5. T he Green metric and the Martin compactiﬁcation. G iv en a probabilit y measure µ on a coun table g roup Γ, one deﬁnes the Martin k ernel K ( x, y ) = K y ( x ) def. = G ( x, y ) G ( e, y ) . By deﬁnition, the Martin c omp actiﬁc ation Γ ∪ ∂ M Γ is the smallest compactiﬁcation of Γ endo w ed with the discrete top o logy suc h that the Martin k ernel con tin uously extends to Γ × (Γ ∪ ∂ M Γ). Then ∂ M Γ is called the Martin b oundary . HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 7 A general theme is to iden tify the Martin b oundary with a geometric b oundary of the group. It w as observ ed in [8] that the Martin compactiﬁcation coincides with the Buse mann compactiﬁcation of (Γ , d G ). W e go one step further by sho wing t ha t the Green metric pro vides a common framew ork for t he identiﬁcation of the Martin boundar y with the b oundary at inﬁnit y o f a h yp erb olic space (cf. [1, 3, 28]). Theorem 1.7. L et Γ b e a c ountable gr oup, µ a symmetric pr ob ability me asur e the supp ort of which gener ates Γ . We assume that the c orr esp onding r and o m walk is tr ansient. If the Gr e en metric is hyp e rb olic, then the Martin b ounda ry c onsists only of m inimal p oints and it is home omorph ic to the hyp erb olic b ounda ry of (Γ , d G ) . In p articular, if Γ is a n o n-elementary hyp erb olic gr oup a nd if d G ∈ D (Γ) , then ∂ M Γ is home om orphic to ∂ Γ . One easily deduces f rom Corollary 1.2: Corollary 1.8. (A. Ancona) L et Γ b e a non - elementary hyp erb olic gr oup, µ a ﬁnitely sup- p orte d pr ob ability me asur e the supp ort of which gener a tes Γ . Then t he Mart in b ounda ry is home om orphic to the hyp erb olic b oundary o f Γ . In § 6.3, we pro vide examples of h yp erb olic groups with random w alks for whic h the Green metric is hyperb olic, but not in the quasi-isometry class of the group, a nd also examples of non-h yp erb olic groups for whic h the Gr een metric is nonetheless h yperb o lic. T hese examples are constructed b y discretising Bro wnian motio ns on Riemannian manifolds (see b elo w). 1.6. B ro wnian motion revisited. Let M b e the univ ersal co v ering of a compact Riemann- ian manifold of nega t ive curv ature with dec k transformat io n group Γ i.e., the action of Γ is isometric, co compact and prop erly discon tin uous. The Br ownian motion ( ξ t ) on M is the diﬀusion pro cess generated by the Laplace-Beltrami op erator. It is kno wn that the Brown ian motion tra jectory almost surely con v erges to some limit p oint ξ ∞ ∈ ∂ M . The law of ξ ∞ is the harmonic measure of the Brownian mo t io n. The notions of rate of escap e and asymptotic en trop y also mak e p erfect sens e in this setting. Reﬁning a metho d of T. Ly ons and D. Sulliv an [40], W. Ballmann and F. Ledrappier con- struct in [3] a random w alk on Γ whic h mirrors the tra jectories o f the Brownian motion and to whic h w e ma y apply our previous results. This enables us to reco v er the follo wing results. Theorem 1.9. L et M b e the universal c o vering of a c omp act Riemannian m anifold of n e gative curvatur e with lo garithmic volume gr owth v . L et d ε b e a visual distanc e on ∂ M . Then dim ν = h M εℓ M wher e h M and ℓ M denote the asymptotic e n tr opy a nd the drift of the Br ownian motion r e- sp e ctively. F urthermor e, h M = ℓ M v if and only if ν is e quivalent to the Hausdorﬀ me asur e of dimension v /ε on ( ∂ M , d ε ) . The ﬁrst result is folklore and explicitely stated b y V. Kaimanov ic h in the in tro duction of [26], but w e kno w of no published pro of. The second statemen t is due to F. Ledrappier [36]. Note that more is kno wn: the equalit y h M = ℓ M v is equiv alent to the equalit y of ν with the canonical c onformal measure on ( ∂ M , d ε ), and this is p ossible only if M is a ra nk 1 symmetric space [37, 5]. 8 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU 1.7. Quasiruled h yp erb olic spaces. As previously mentioned, S. Blac h ` ere and S. Broﬀerio pro v ed that, for ﬁnitely supp orted la ws, the Green metric d G is quasi-isometric to t he w ord metric. But since d G is deﬁned only on a coun table set, it is unlik ely to b e the restriction of a prop er geo desic metric (whic h would ha v e guaranteed the h yp erb olicity of (Γ , d G )). Therefore, the pro of of Theorem 1.1 requires the understanding of whic h metric spaces among the quasi- isometry class of a giv en geo desic h yp erb o lic space are a lso h yp erb o lic. F or this, w e coin the notion of a quasiruler: a τ -quasiruler is a quasigeo desic g : R → X suc h that, for a ny s < t < u , d ( g ( s ) , g ( t )) + d ( g ( t ) , g ( u )) − d ( g ( s ) , g ( u )) ≤ 2 τ . A metric space will b e q uasi rule d if constants ( λ, c, τ ) exist so that the space is ( λ, c ) - quasi- geo desic and if ev ery ( λ, c )-quasigeo desic is a τ -quasiruler. W e refer to the App endix for details on the deﬁnitions and prop erties of quasigeo desics a nd quasiruled spaces. W e prov e the fo llowing c haracterisation of hy p erb olicity , interes ting in its o wn righ t. Theorem 1.10. L et X b e a ge o desic hyp erb olic me tric sp ac e, and ϕ : X → Y a quasi-isometry, wher e Y is a metric sp ac e. Then Y is hyp erb olic if and only if it is quasirule d. Theorem 1.10 will b e used to pro v e that the h yperb olicit y of d G is equiv a len t to condition (1) in Theorem 1.1. W e complete this discussion b y exhibiting f or any hy p erb olic group, a non-h yp erb olic left-in v a rian t me tric in its quasi-isometry class (cf. Prop osition A.11). 1.8. F uch sian groups with cusps. W e pro vide an alternative pro of based on the Green metric of a theorem due to Y. G uiv arc’h and Y. Le Jan ab out random w alks on F uc hsian groups, see the last corollary of [20]. Theorem 1.11. (Y. Guiv arc’h & Y. Le Jan) L et Γ b e a d iscr ete sub gr oup of P S L 2 ( R ) such that the quotient sp ac e H 2 / Γ is not c omp act but has ﬁnite volume. L et ν 1 b e the harmonic me asur e on S 1 given by a symmetric law µ with ﬁn i te supp ort. ( Almost any tr aje ctory of the r andom walk c onver ges to a p oint in ∂ H 2 = S 1 and ν 1 is the law of this limit p oint.) Then ν 1 is singular with r esp e ct to the L eb es gue me asur e on S 1 . Note that it f o llo ws fro m T heorem 1.5 that ν 1 is singular with resp ect to the Leb esgue measure if and only if its dimension is less than 1. This theorem was originally deriv ed from results on winding num b ers of the geo desic ﬂo w, see [20] and [2 1]. A more recen t pro of based on ergo dic prop erties of smo oth g roup actions on S 1 w as obtained b y B. D ero in, V. Kleptsyn and A. Na v as in [16]. It applies to random walks with a ﬁnite ﬁrst momen t. W e shall see how Theorem 1.11 can also b e deduced from the h yperb olicit y of the Green metric through a rather straighforw a r d argumen t. W e only conside r the symmetric and ﬁnite supp ort case ev en though it w ould also work if the random w alk has a ﬁrst ﬁnite momen t and if d G ∈ D (Γ). W e thank B. Deroin, Y. Guiv arc’h and Y. Le Jan for enligh tening explanatio ns on their theorem. 1.9. Out line of the paper. In Section 2, w e recall the main facts o n h yp erb olic groups whic h will b e used in the pap er. In Section 3 , w e recall the construction of random w alks, discuss some of their prop erties a nd in tro duce the Green metric. W e also pro ve Theorem 1.7 and Theorem 1.1. W e then dra w some consequenc es on the harmonic measure and the random walk . The follow ing Section 4 deals with the pro o f of Theorem 1.3. In Section 5, HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 9 w e deal with Theorem 1.5 and its corolla ry and w e conclude with the pro of o f Theorem 1.11. Finally , Theorem 1.9 is pro v ed in Section 6. The app endices are dev ot ed to quasiruled spaces. W e pro v e Theorem 1.10 in App endix A, and we sho w that quasiruled spaces retain most prop erties of geo desic hyperb olic spaces : in App endix B, w e sho w that the approximation of ﬁnite conﬁgurations by trees s till hold, and we explain wh y M. Co ornaert’s theorem o n quasiconformal measures remains v alid in this setting. 1.10. Notation. A distance in a metric space will b e denoted either b y d ( · , · ) or | · − · | . If a and b ar e p ositiv e, a . b means that there is a univ ersal p ositive constant u suc h that a ≤ ub . W e will write a ≍ b when bo t h a . b and b . a hold. Throughout the article, dep endance of a constan t on structural parameters of the space will not be notiﬁed unless needed. Some times, it will b e con v enien t to use Landau’s notation O ( · ). 2. Hyp erbolicity in metric sp aces Let ( X , d ) b e a metric space. It is said to b e pr op er if closed balls of ﬁnite radius are compact. A ge o desic curve (r esp. r ay, s e gment) is a curv e isometric to R (resp. R + , a compact in terv al of R ). T he space X is said to b e ge o desic if ev ery pair of p oints can b e joined b y a geo desic segmen t. Giv en three p oin ts x, y , w ∈ X , one deﬁnes the Gromov inner pro duct as follow s: ( x | y ) w def. = (1 / 2) {| x − w | + | y − w | − | x − y |} . Deﬁnition. A metric space ( X , d ) is δ -hyp erb olic ( δ ≥ 0) if, for an y w , x, y , z ∈ X , the follo wing ult r a metric type inequalit y holds ( y | z ) w ≥ min { ( x | y ) w , ( x | z ) w } − δ . W e shall write ( ·|· ) w = ( · | · ) when the choice of w is clear from the con text. Hyp erb olicit y is a lar g e-scale pr o p ert y of the space. T o capture this info r ma t ion, one deﬁnes the notio n of quas i-isometry . Deﬁnition. Let X , Y b e tw o metric spaces and λ ≥ 1, c ≥ 0 tw o constan ts. A map f : X → Y is a ( λ, c ) -quasi - i s ometric emb e dding if, for a n y x, x ′ ∈ X , w e hav e 1 λ | x − x ′ | − c ≤ | f ( x ) − f ( x ′ ) | ≤ λ | x − x ′ | + c . The map f is a ( λ , c ) -quasi-isometry if, in addition, there exist a quasi-isometric em b edding g : Y → X and a constant C such that | g ◦ f ( x ) − x | ≤ C fo r an y x ∈ X . Equiv alen tly , f is a quasi-isometry if it is a quasi-isometric em bedding suc h that Y is con tained in a C - neigh b orho o d of f ( X ). W e then say that f is C -c ob ounde d . In the sequel, w e will alw a ys c hoo se the constan ts so that that a ( λ, c )-quasi-isometry is c -cob ounded. Deﬁnition. A quasige o desic curve (r e sp. r ay, se gment) is the imag e of R (resp. R + , a compact interv al of R ) b y a quasi-isometric em b edding. 10 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU In a geodesic h yperb olic metric space ( X, d ), quasigeo desics alw ays shado w genuin e geo desics i.e., giv en a ( λ, c ) - quasigeo desic q , there is a geo desic g suc h that d H ( g , q ) ≤ K , where d H denotes t he Hausdorﬀ distance, and K only depends on δ , λ and c [22, Th. 5.6]. Compactiﬁcation. Let X b e a prop er hyperb olic space, and w ∈ X a base p oin t. A sequenc e ( x n ) tends to inﬁnit y if, b y deﬁnition, ( x n | x m ) → ∞ as m, n → ∞ . The visual or hyp erb olic b oundary ∂ X of X is the set of sequences whic h tend to inﬁnity modulo the equiv alence relation deﬁned b y: ( x n ) ∼ ( y n ) if ( x n | y n ) → ∞ . One may also extend the Gromov inner pro duct to p oints at inﬁnit y in suc h a w a y that the inequality ( y | z ) ≥ min { ( x | y ) , ( x | z ) } − δ , no w holds for an y po in ts w , x, y , z ∈ X ∪ ∂ X . F or each ε > 0 small enough, there exists a so- called visual m e tric d ε on ∂ X i.e whic h satisﬁes for an y a, b ∈ ∂ X : d ε ( a, b ) ≍ e − ε ( a | b ) . W e shall use t he nota tion B ε ( a, r ) to denote the ba ll in the space ( ∂ X , d ε ) with cente r a and radius r . W e refer to [22] for the details (c hap. 6 and 7). Busemann functions. Let us assume that ( X , d ) is a hy p erb olic space. Let a ∈ ∂ X , x, y ∈ X . The function β a ( x, y ) def. = sup  lim sup n →∞ [ d ( x, a n ) − d ( y , a n )]  , where the suprem um is tak en ov er all seque nces ( a n ) n in X whic h tends to a , is called the Busemann function at the p oint a . Shado ws. Let R > 0 and x ∈ X . The shadow ℧ ( x, R ) is the set of points a ∈ ∂ X suc h that ( a | x ) w ≥ d ( w , x ) − R . Appro ximating ﬁnitely many p oin ts b y a tree (cf. Theorem B.1) yields: Prop osition 2.1. L et ( X , d ) b e a hyp erb olic sp ac e. F or any τ ≥ 0 , ther e exist p ositive c on- stants C , R 0 such that for any R > R 0 , a ∈ ∂ X an d x ∈ X such that ( w | a ) x ≤ τ , B ε  a, 1 C e Rε e − ε | w − x |  ⊂ ℧ ( x, R ) ⊂ B ε  a, C e Rε e − ε | w − x |  . Shado ws will enable us to con trol measures on the b oundary of a hyperb olic group, see the lemma of the shado w in the nex t paragraph. 2.1. H yp erb olic gro ups. Let X be a h yp erb olic pro p er metric space and Γ a subgroup of isometries whic h acts prop erly discon tin uously on X i.e., for an y compact sets K and L , the n um ber of group elemen ts γ ∈ Γ suc h tha t γ ( K ) ∩ L 6 = ∅ is ﬁnite. F or a ny p oin t x ∈ X , its orbit Γ( x ) accum ulates only on the b o undar y ∂ X , and its set of a ccum ula tion p oints turns out to be independent o f the c hoice of x ; b y deﬁnition, Γ( x ) ∩ ∂ X is the li mit set Λ(Γ) of Γ. An action of a group Γ on a metric space is said to be ge ometric if (1) each elemen t acts b y isome try; (2) the action is prop erly discontin uous; HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 11 (3) the action is co compact. F or example, if Γ is a ﬁnitely generated group, S is a ﬁnite symmetric set of generators, one ma y consider the Cayley graph X asso ciated with S : the set of v ertices are the elemen ts of the group, and pairs ( γ , γ ′ ) ∈ Γ × Γ deﬁne an edge if γ − 1 γ ′ ∈ S . Endo wing X with the metric whic h mak es eac h edge isometric to the segmen t [0 , 1 ] deﬁnes the wor d metric asso ciate d with S . It turns X in to a geo desic prop er metric space on which Γ acts geometrically b y left-translation. W e recall ˇ Sv arc-Milnor’s lemma whic h pro vides a sort of con ve rse statemen t , see [22]: Lemma 2.2. L et X b e a ge o desic pr op er metric sp ac e, and Γ a gr oup which acts ge ometric al ly on X . Then Γ is ﬁnitely gener ate d and X is quasi - i s ometric to any lo c al ly ﬁnite Cayley gr ap h of Γ . A group Γ is hyp erb olic if it a cts geometrically on a geo desic prop er h yperb o lic metric space (e.g. a lo cally ﬁnite Cayle y g r aph). In this case, one has Λ( Γ ) = ∂ X . Then ˇ Sv arc-Milnor’s lemma ab o v e implies that Γ is ﬁnitely generated. W e will say that a metric space ( X , d ) is quasi-isome tric to the g roup Γ if it is quasi-isometric to a locally ﬁnite Cay ley graph of Γ. Let Γ b e a h yp erb olic gro up geometrically acting o n ( X , d ). The action of Γ extends to the b oundary . Busemann functions, visual metrics and the action of Γ are related b y the fo llo wing prop erty : for an y a ∈ ∂ X and any γ ∈ Γ, there exists a neigh b orho o d V of a suc h that, for an y b, c ∈ V , d ε ( γ ( b ) , γ ( c )) ≍ L γ ( a ) d ε ( b, c ) where L γ ( a ) def. = e εβ a ( w, γ − 1 ( w )) . Moreov er, Γ a lso acts on measures on ∂ X through the rule γ ∗ ρ ( A ) def. = ρ ( γ A ). A hy p erb olic group is said to b e elemen tary if it is ﬁnite or quasi-isometric to Z . W e will only b e dealing with non- elemen tary h yperb olic groups. 2.2. Quasiconformal measures. W e no w assume that Γ is a hyperb o lic group acting on a prop er quasiruled h yp erb olic metric space ( X, d ). The nex t theorem summarizes the ma in prop erties of quasiconformal measures on the b oundary of X . I t was prov ed b y M . Co ornaert in [15] in the con text of geo desic spaces. W e stat e here a more general v ersion to co v er the case d ∈ D ( Γ). W e justify the v a lidit y of this generalisation at the end o f the app endix. W e refer to Section 4 for the deﬁnitions of the Hausdorﬀ measure and dimension. Theorem 2.3. L et ( X , d ) b e a pr o p er quasirule d hyp e rb olic sp ac e endowe d with a ge ometric action of a non-ele m entary hyp erb olic gr oup Γ . F or any smal l enough ε > 0 , we have 0 < dim H ( ∂ X , d ε ) < ∞ and v def. = lim sup 1 R log |{ Γ( w ) ∩ B ( w , R ) } | = ε · dim H ( ∂ X , d ε ) . L et ρ b e the Hausdorﬀ me asur e on ∂ X of dimension α def. = v /ε ; (i) ρ is Ahlfors-r e gular of dimensio n α i.e., for any a ∈ ∂ X , fo r any r ∈ ( 0 , diam ∂ X ) , ρ ( B ε ( a, r )) ≍ r α . In p articular, 0 < ρ ( ∂ X ) < ∞ . 12 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU (ii) ρ is a quasic onform al me asur e i.e., for any isometry γ we have ρ ≪ γ ∗ ρ ≪ ρ and dγ ∗ ρ dρ ≍ ( L γ ) α ρ a.e. . (iii) The action of Γ is er go dic for ρ i.e., for any Γ -invariant Bor elian B of ∂ X , ρ ( B ) = 0 or ρ ( ∂ X \ B ) = 0 . Mor e o v er, if ρ ′ is another Γ -quasic onformal me asur e, then ρ ≪ ρ ′ ≪ ρ and dρ dρ ′ ≍ 1 a . e. and |{ Γ( w ) ∩ B ( w , R ) }| ≍ e vR . The class of measures th us deﬁned on ∂ X is called the Patterson-Sul livan class. It do es no t dep end on the choice of the parameter ε but it do es dep end on the metric d . The study of quasiconformal measures yie lds the followin g k ey estimate [1 5]: Lemma 2.4. (Lemma of the shado w) Under the assumptions of The or em 2 .3, ther e exists R 0 , such that if R > R 0 , then, for any x ∈ X , ρ ( ℧ ( x, R )) ≍ e − vd ( w ,x ) wher e the implicit c o n stants do not dep end on x . 3. Random w alks and Gree n metrics for hype rbolic groups Let Γ b e a h yp erb olic gr oup, and let us consider the set D (Γ) of left-in v arian t h yp erb olic metrics on Γ whic h are quasi-isometric to Γ. W e ﬁx suc h a metric ( X , d ) ∈ D (Γ) with a base p oint w ∈ X , and we consider a symmetric probability measure µ on Γ with ﬁnite ﬁrst momen t i.e. X γ ∈ Γ µ ( γ ) d ( w , γ ( w )) < ∞ . The random w alk ( Z n ) n starting from t he neutral elemen t e asso ciated with µ is deﬁned b y the recursion relations: Z 0 = e ; Z n +1 = Z n · X n +1 , where ( X n ) is a seque nce o f indep enden t and iden tically distributed r a ndom v ariables of la w µ . Thu s, for eac h n , Z n is a random v ariable taking its v a lues in Γ. W e use the nota tion Z n ( w ) for the image of the base p oin t w ∈ X b y Z n . The r ate o f esc ap e , or drift of the r a ndom w alk Z n ( w ) is the num b er ℓ deﬁned as ℓ def. = lim n d ( w , Z n ( w )) n , where the limit exists almost surely and in L 1 b y the sub-additive ergo dic The orem (J. Kingman [33], Y. Derriennic [17]). If Γ is elemen tary , then its b oundary is either empt y or ﬁnite. In either case, t here is no in terest in lo oking at pro p erties at the b oundary . W e will assu me f r o m no w on that Γ is non- elemen tar y . In particular, Γ is non-amenable so not only is the random w alk alw a ys tra nsient, ℓ is also p ositive (cf. [30, § 7.3]). There are diﬀeren t w ays to pro v e that almost a n y tra jectory of t he random w alk has a limit p oin t Z ∞ ( w ) ∈ ∂ X . W e recall b elow a theorem by V. Kaimanovic h (cf. Theorem 7.3 in [30] HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 13 and § 7.4 therein) since it contains some information on the w a y ( Z n ( w )) actually t ends to Z ∞ ( w ) that will be us ed later. Theorem 3.1. ( V. Kaimano vic h) . L e t Γ b e a n o n-elementary hyp erb olic gr o up and ( X , d ) ∈ D (Γ) , and let us c onsider a symmetric pr ob ability me asur e µ with ﬁnite ﬁrst mo m ent the supp ort of which gener a tes Γ . Then ( Z n ( w )) almost sur ely c o n ver ges to a p oint Z ∞ ( w ) on the b oundary. F or any a ∈ ∂ X , we cho os e a quasige o desic [ w , a ) fr om w to a in a me asur able way. F or any n , ther e is a me as ur able map π n fr om ∂ X to X such that π n ( a ) ∈ [ w , a ) , an d, for almost any tr aje ctory of the r andom walk, (2) lim n →∞ | Z n ( w ) − π n ( Z ∞ ( w )) | n = 0 . The a ctual result w as pro v ed for g eo desic metrics d . Once prov ed in a locally ﬁnite Ca yley graph, one ma y then use a quasi-isometry to get the stat emen t in this generalit y . The estimate (2) will b e impro ve d in Corollary 3.9 under the condition t hat d G b elongs to D (Γ). The harmonic me asur e ν is then the la w of Z ∞ ( w ) i.e., it is the probabilit y measure o n ∂ X suc h that ν ( A ) is t he probability that Z ∞ ( w ) belongs to the set A . More g enerally , w e let ν γ b e the harmonic measure for the random w alk started at the p oint γ ( w ), γ ∈ Γ i.e. the law of γ ( Z ∞ ( w )). Comparing with the action o f Γ on ∂ X , w e se e t ha t γ ∗ ν = ν γ − 1 . 3.1. T he Green metric. Let Γ be a coun table group and µ a symmetric law the supp ort of whic h generates Γ. F or x, y ∈ Γ, w e deﬁne F ( x, y ) as the probabilit y that a random w a lk starting from x hits y in ﬁnite time i.e., the probabilit y there is some n such that xZ n = y . S. Blac h ` ere and S. Broﬀerio [7] hav e deﬁned the Green metric b y d G ( x, y ) def. = − log F ( x, y ) . The Mar ko v prop erty implies tha t F and the Green function G satisfy G ( x, y ) = F ( x, y ) G ( y , y ) . Since G ( y , y ) = G ( e, e ), w e then get that F ( x, y ) = G ( x, y ) G ( e, e ) i.e. F and G only diﬀer b y a m ultiplicative contan t and d G ( x, y ) = log G ( e, e ) − log G ( x, y ) . This f unction d G is know n to b e a left-inv ariant metric o n Γ (see [7, 8] for details). W e end this short in tro duction to the G reen metric with the follo wing folklore prop erty . Lemma 3.2. L et µ b e a symmetric pr ob ability me asur e o n Γ which deﬁne s a tr ansient r andom walk. Then ( Γ , d G ) is a pr op er metric sp ac e i.e., b al ls of ﬁnite r adius ar e ﬁ n ite. Proo f. It is enough t o pro ve that G ( e, x ) tends to 0 as x leav es any ﬁnite set. 14 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU Fix n ≥ 1; b y deﬁnition of conv olution and by the Cauc hy -Sc h w a rz inequalit y , µ 2 n ( x ) = X y ∈ Γ µ n ( y ) µ n ( y − 1 x ) ≤ s X y ∈ Γ µ n ( y ) 2 s X y ∈ Γ µ n ( y − 1 x ) 2 . Since we are su mming o v er the same set, it follo ws that X y ∈ Γ µ n ( y ) 2 = X y ∈ Γ µ n ( y − 1 x ) 2 and the sym metry of µ implies that X y ∈ Γ µ n ( y ) 2 = X y ∈ Γ µ n ( y ) µ n ( y − 1 ) = µ 2 n ( e ) . Therefore, µ 2 n ( x ) ≤ µ 2 n ( e ). Similarly , µ 2 n +1 ( x ) = X y ∈ Γ µ ( y ) µ 2 n ( y − 1 x ) ≤ X y ∈ Γ µ ( y ) µ 2 n ( e ) ≤ µ 2 n ( e ) . Since the w alk is transien t, it follows that G ( e, e ) is ﬁnite, so, giv en ε > 0 , there is some k ≥ 1 suc h that X n ≥ k µ 2 n ( e ) ≤ X n ≥ 2 k µ n ( e ) ≤ ε . On the other hand, since µ n is a probabilit y measure for all n , there is some ﬁnite subset K of Γ suc h that, for all n ∈ { 0 , . . . , 2 k − 1 } , µ n ( K ) ≥ 1 − ε / (2 k ) . The refore, if x 6∈ K , then G ( e, x ) = X 0 ≤ n< 2 k µ n ( x ) + X n ≥ 2 k µ n ( x ) ≤ X 0 ≤ n< 2 k µ n (Γ \ K ) + 2 X n ≥ k µ 2 n ( e ) ≤ ε + 2 ε . The lemma follo ws. 3.2. T he Martin b oundary. Let Γ b e a coun table group and µ b e a symme tric probabilit y measure on Γ . W e assume that the supp ort of µ generates Γ and that the corresponding random w alk is transien t. A non- negativ e function h on Γ is µ -harmonic (harmonic f or short) if, fo r all x ∈ Γ, h ( x ) = X y ∈ Γ h ( y ) µ ( x − 1 y ) . A p ositiv e harmo nic function h is min imal if an y other p ositive harmonic function v smaller than h is prop or t io nal to h . The Martin k ernel is deﬁned for all ( x, y ) ∈ Γ × Γ by K ( x, y ) def. = G ( x, y ) G ( e, y ) = F ( x, y ) F ( e, y ) . W e endo w Γ with the discrete top ology . Let us brieﬂy recall the construction of the Martin b oundary ∂ M Γ: let Ψ : Γ → C (Γ ) b e deﬁned by y 7− → K y = K ( · , y ). Here C (Γ ) is the space of real- v alued functions deﬁned on Γ endow ed w ith the top ology of p oin t wise conv ergence. It t urns out that Ψ is injectiv e and th us w e ma y iden tify Γ with its image. The closure of Ψ(Γ) is compact in C (Γ) and, by deﬁnition, ∂ M Γ = Ψ(Γ) \ Ψ(Γ) is the Martin boundar y . In the compact space Γ ∪ ∂ M Γ, for an y initial p oint x , the random walk Z n ( x ) a lmost surely con v erges to some random v ariable Z ∞ ( x ) ∈ ∂ M Γ (see f o r instance E. D ynkin [18], A. Ancona [1] or W. W o ess [49]). HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 15 T o ev ery p oint ξ ∈ ∂ M Γ correspo nds a positive harmonic function K ξ . Ev ery minimal function arises in this w ay: if h is minimal, then there are a constan t c > 0 and ξ ∈ ∂ M Γ suc h that h = cK ξ . W e denote b y ∂ m Γ t he subs et of ∂ M Γ consisting of (no rmalised) minimal p ositiv e harmonic functions. Cho quet’s in tegral repres en tation implies that, f o r an y p ositiv e harmonic function h , there is a unique probability measure κ h on ∂ m Γ suc h that h = Z K ξ dκ h ( ξ ) . W e will also use L. Na ¨ ım’s k ernel Θ on Γ × Γ deﬁned b y Θ( x, y ) def. = G ( x, y ) G ( e, x ) G ( e, y ) = K y ( x ) G ( e, x ) . As the Martin kerne l, Na ¨ ım’s kerne l admits a contin uous extens ion to Γ × (Γ ∪ ∂ M Γ). In terms of the Green metric, one gets (3) log Θ( x, y ) = 2( x | y ) G e − log G ( e, e ) , where ( x | y ) G e denotes the Gromo v pro duct with respect to the G r een metric. See [43] for prop erties o f this k ernel. W e shall from no w on assume that the Green metric d G is h yp erb olic. Then it has a visual b oundary that we denote b y ∂ G Γ. W e may a lso compute the Busemann function in the metric d G , sa y β G a . Sending y to some p oin t a ∈ ∂ G Γ in the equation d G ( e, y ) − d G ( x, y ) = log K ( x, y ), w e get that β G a ( e, x ) = log K a ( x ). W e no w start preparing the pro of of Theorem 1.7 in the nex t lemma and prop osition. W e deﬁne an eq uiv alence relation ∼ M on ∂ M Γ: sa y that ξ ∼ M ζ if there exists a constan t C ≥ 1 suc h that 1 C ≤ K ξ K ζ ≤ C . Giv en ξ ∈ ∂ M Γ, w e denote b y M ( ξ ) the class of ξ . W e ﬁrst deriv e some prop erties of this equiv alence relation: Lemma 3.3. (i) Ther e exists a c onstant E ≥ 1 such that for al l se quenc es ( x n ) and ( y n ) in Γ c onver ging to ξ and ζ in ∂ M Γ r esp e ctively and such that Θ( x n , y n ) tends to i n ﬁnity, then 1 E ≤ K ξ K ζ ≤ E ; in p articular, ξ ∼ M ζ . (ii) F or any ξ ∈ ∂ M Γ , ther e is some ζ ∈ M ( ξ ) and a se quenc e ( y n ) in Γ w hich tends to some p o i n t a ∈ ∂ G Γ in the sense of Gr omov, to ζ ∈ ∂ M Γ in the sense of Martin and such that Θ( y n , ξ ) tends to inﬁnity. (iii) L et ξ , ζ ∈ ∂ M Γ . If ζ / ∈ M ( ξ ) , then ther e is a neighb orh o o d V ( ζ ) of ζ in Γ and a c onstant M such that K ξ ( x ) ≤ M G ( e, x ) for any x ∈ V ( ζ ) . Proo f. (i) Fix z ∈ Γ and n large enough so that ( x n | y n ) G e ≫ d G ( e, z ); we consider the approxim ate tree T asso ciated with F = { e, z , x n , y n } and the (1 , C )-quasi-isometry ϕ : ( F , d G ) → ( T , d T ) (cf. Theorem B.1). 16 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU On t he tree T ,w e ha ve | d T ( ϕ ( e ) , ϕ ( x n )) − d T ( ϕ ( z ) , ϕ ( x n )) | = | d T ( ϕ ( e ) , ϕ ( y n )) − d T ( ϕ ( z ) , ϕ ( y n )) | , so that | ( d G ( e, x n ) − d G ( z , x n )) − ( d G ( e, y n ) − d G ( z , y n )) | ≤ 2 C . In terms of the Martin k ernel, | log K x n ( z ) − log K y n ( z ) | ≤ 2 C . Letting n go to inﬁnit y yields the result. (ii) Let ( y n ) b e a sequence suc h that lim K ξ ( y n ) = sup K ξ . Since K ξ is ha r mo nic, the maximum principle implies that ( y n ) leav es any compact set. But the walk is symmetric and t r a nsien t so Lemm a 3.2 implies that G ( e, y n ) tends to 0. F urthermore, for n la rge enoug h, K ξ ( y n ) ≥ K ξ ( e ) = 1 , so that Θ( y n , ξ ) ≥ 1 G ( e, y n ) → ∞ . Let ( x n ) b e a sequen ce in Γ whic h tends to ξ . F or any n , there is some m suc h that | K ξ ( y n ) − K x m ( y n ) | ≤ G ( e, y n ) . It follows that Θ( y n , x m ) ≥ Θ( y n , ξ ) − | K ξ ( y n ) − K x m ( y n ) | G ( e, y n ) ≥ Θ( y n , ξ ) − 1 . Therefore, applying part (i) of the lemma, w e see that an y limit p oint of ( y n ) in ∂ M Γ b elongs to M ( ξ ). Moreo v er, for an y suc h limit p o in t ζ ∈ ∂ M Γ, we get that Θ( y n , ζ ) ≥ 1 E Θ( y n , ξ ) . Applying the same argument as ab o v e, w e see that, fo r an y M > 0, there is some n and m n suc h that, if m ≥ m n then Θ( y n , y m ) ≥ M − 1 . F rom (3) we conclude, using a diagonal pro cedure, tha t there exist a subsequence ( n k ) suc h that ( y n k ) tends to inﬁnit y in the Gromo v top ology . (iii) Since ζ / ∈ M ( ξ ), there is a neigh b orho o d V ( ζ ) and a constan t M suc h that Θ( x, ξ ) ≤ M for all x ∈ V ( ζ ). Otherwise, w e would ﬁnd y n → ζ with Θ( y n , ξ ) going to inﬁnit y , and the argumen t ab o v e w ould imply ζ ∈ M ( ξ ). Therefore, K ξ ( x ) ≤ M G ( e, x ) . Prop osition 3.4. Every Martin p oin t is minimal. HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 17 Proo f. W e observ e that if K ξ is minimal, then M ( ξ ) = { ξ } . Indeed, if ζ ∈ M ( ξ ), then K ξ ≥ K ξ − 1 C K ζ ≥ 0 for some constan t C ≥ 1. The minimality o f K ξ implies that K ξ and K ζ are prop ortional and, since their v alue a t e is 1, K ξ = K ζ i.e., ξ = ζ . Let ξ ∈ ∂ M Γ. There is a unique probabilit y me asure κ ξ on ∂ m Γ suc h that K ξ = Z K ζ dκ ξ ( ζ ) . By F ato u-Do o b-Na ¨ ım Theorem, for κ ξ -almost eve ry ζ , t he ratio G ( e, x ) /K ξ ( x ) tends to 0 as x tends to ζ in the ﬁne top ology [1, Thm . I I.1.8]. F rom Lemma 3.3 (iii), it follows that κ ξ is supp orted by M ( ξ ). In particular, M ( ξ ) con tains a minimal p o in t. Proo f of Theorem 1.7. Since ev ery Martin p oint is minimal, Lemma 3.3, ( ii) , implies that for ev ery ξ ∈ ∂ M Γ, t here is some se quence ( x n ) in Γ whic h tends to ξ in the Martin top ology and to some p oint a in the hy p erb olic b oundary as w ell. Let us prov e that the p oin t a do es not dep end o n the sequence. If ( y n ) is anot her sequence tending to ξ , then lim sup n,m →∞ Θ( x n , y m ) = ∞ b ecause Θ( ξ , x n ) tends to inﬁnity . Therefore, there is a subsequence of ( y n ) whic h tends t o a in the Gromov top ology . Since we hav e only one accum ulation p oint, it follo ws tha t a is w ell-deﬁned. This deﬁnes a map φ : ∂ M Γ → ∂ G Γ. No w, if ( x n ) tends to a in the Gro mov top o logy , then it has only one accum ulation p o int in the Martin b oundary a s w ell by Lemma 3.3, (i). So the map φ is injectiv e. The surjectivit y follo ws from the compactness of ∂ M Γ. T o conclu de the pro o f, it is enough to pro v e the con tinuit y of φ since ∂ M Γ is compact. Let M > 0 a nd ξ ∈ ∂ M Γ b e giv en. W e consider a sequence ( x n ) whic h tends to ξ as in Lemma 3.3. Let C b e the constan t giv en b y Theorem B.1 fo r 4 p oin ts. W e pic k n large enough so that ( x n | φ ( ξ )) G e ≥ M + 2 C + log 2. Let A = min { K ξ ( x ) , x ∈ B G ( e, d G ( x n , e )) } . Let ζ ∈ ∂ M Γ suc h that | K ξ − K ζ | ≤ ( A/ 2) o n B G ( e, d G ( x n , e )). It follo ws that 1 / 2 ≤ K ζ K ξ ≤ 3 / 2 . Appro ximating { e, x n , φ ( ξ ) , φ ( ζ ) } by a tree, w e conclude that ( φ ( ξ ) | φ ( ζ )) G e ≥ M , pro ving the contin uity of φ . 3.3. H yp erb olicity of t he Green metric. W e start with a characterisation of the hy p er- b olicit y o f the Green metric in the quasi-isometry class of the group. Prop osition 3.5. L e t Γ b e a non -elementary hyp erb olic gr oup and µ a symme tric pr ob ability me asur e with Gr e en function G . We ﬁx a ﬁnite gene r ating set S and c onside r the asso ciate d wor d metric d w . The Gr e en metric d G is quasi-iso m etric to d w and hyp erb olic i f and on ly if the fol lowing two c ond itions ar e satisﬁe d. 18 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU (ED) Th e r e ar e p ositive c onstants C 1 and c 1 such that, for al l γ ∈ Γ , G ( e, γ ) ≤ C 1 e − c 1 d w ( e,γ ) (QR) F or any r ≥ 0 , ther e exists a p ositive c onstant C ( r ) such that G ( e, γ ) ≤ C ( r ) G ( e, γ ′ ) G ( γ ′ , γ ) whenever γ , γ ′ ∈ Γ and γ ′ is at distanc e at most r fr om a d w -ge o desic se gment b etwe en e and γ . Remark. Ev en thoug h hy p erb olicity is an inv ariant prop erty under quasi-isometries b et w een geo desic metric spaces, this is not the case when w e do not a ssume the spaces to b e geo desic (see the appendix). Proo f. W e ﬁrst assume that d G ∈ D (Γ ) . The quasi-isometry pro p ert y implies that condition (ED) holds. The second condition (QR) follo ws from Theorem A.1. Indeed, since d G is h yp erb olic and quasi-isometric to a word distance, then (Γ , d G ) is quasir- uled. This is suﬃcien t to ensu re that condition (QR) holds for r = 0. Th e g eneral case r ≥ 0 follo ws: let y b e the closest p oin t to γ ′ on a geo desic b etw een e and γ and note that d G ( e, γ ′ ) + d G ( γ ′ , γ ) ≤ d G ( e, y ) + d G ( y , γ ) + 2 d G ( y , γ ′ ) ≤ log C (0) + d G ( e, γ ) + 2 d G ( y , γ ′ ) . Th us one may c ho ose C ( r ) = C (0) exp( 2 c ) where c = sup d G ( y , γ ′ ) for all pair y , γ ′ at distance less than r . This last sup is ﬁnite b ecause d G is quasi-isometric to a w o rd metric. F or the con v erse, w e assume that both conditions (ED) and (QR) hold and let C = max { d G ( e, s ) , s ∈ S } . F o r an y γ ∈ Γ, w e consider a d w -geo desic { γ j } joining e to γ . It follo ws that d G ( e, γ ) ≤ X j d G ( γ j , γ j +1 ) ≤ C d w ( e, γ ) . F rom (ED ), w e obtain d G ( e, γ ) ≥ c 1 d w ( e, γ ) − log C 1 . Since b ot h metrics a re left-inv ariant, it follo ws that d w and d G are quasi-isometric. Condition (QR) implies that d w -geo desics are not only quasigeo desics f or d G , but also quasir- ulers, cf. App endix A. Indeed, since the tw o functions F and G only diﬀer by a m ultiplicativ e factor, condition (QR) implies tha t there is a constan t τ such that, fo r an y d w -geo desic segmen t [ γ 1 , γ 2 ] and an y γ ∈ [ γ 1 , γ 2 ], we ha v e d G ( γ 1 , γ ) + d G ( γ , γ 2 ) ≤ 2 τ + d G ( γ 1 , γ 2 ) . Theorem A.1 , (iii) implies (i), implies that (Γ , d G ) is a h yperb olic space. T o prov e the ﬁrst statemen t of Theorem 1.1, it is now enough to establish the follo wing lemma. Lemma 3.6. L et Γ b e a non-elemen tary hyp erb olic gr oup, and µ a symm etric pr ob ability me asur e w ith ﬁnite exp onential momen t. Then c ondition (ED) holds. When µ is ﬁnitely supp orted, the lemma w a s prov ed by S. Blac h ` ere and S. Br o ﬀerio using the Carne-V a rop oulos es timate [7]. Proo f. Let us ﬁx a w ord me tric d w induced by a ﬁnite g enerating set S , so that d w ∈ D (Γ). HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 19 Since Γ is non-amenable, Kesten’s criterion implies that there are p ositiv e constan ts C and a suc h that (4) ∀ γ ∈ Γ , µ n ( γ ) ≤ µ n ( e ) ≤ C e − an . F or a pro of, s ee [49 , Cor. 12.5]. W e assume that E [exp λd w ( e, Z 1 )] = E < ∞ for a giv en λ > 0. F or any b > 0, it fo llo ws from the exponential Tc heb yc hev inequalit y that P  sup 1 ≤ k ≤ n d w ( e, Z k ) ≥ nb  ≤ e − λbn E  exp  λ sup 1 ≤ k ≤ n d w ( e, Z k )  . But then, for k ≤ n , d w ( e, Z k ) ≤ X 1 ≤ j ≤ n − 1 d w ( Z j , Z j +1 ) = X 1 ≤ j ≤ n − 1 d w ( e, Z − 1 j Z j +1 ) . The incremen ts ( Z − 1 j Z j +1 ) are independen t random v ariables and all f o llo w the same law as Z 1 . Therefore (5) P  sup 1 ≤ k ≤ n d w ( e, Z k ) ≥ nb  ≤ e − λbn E n = e ( − λb +log E ) n . W e choose b large enough so tha t c def. = − λb + log E < 0. W e hav e G ( e, γ ) = X n µ n ( γ ) = X 1 ≤ k ≤ | γ | /b µ k ( γ ) + X k > | γ | /b µ k ( γ ) , where we ha v e set | γ | = d w ( e, γ ) . The estimates (5) and (4 ) resp ectiv ely imply that X 1 ≤ k ≤ | γ | /b µ k ( γ ) ≤ | γ | b sup 1 ≤ k ≤ | γ | /b µ k ( γ ) ≤ | γ | b P [ ∃ k ≤ | γ | /b s.t. Z k = γ ] ≤ | γ | b P " sup 1 ≤ k ≤ | γ | /b d w ( e, Z k ) ≥ | γ | # . | γ | e − c | γ | and X k > | γ | /b µ k ( γ ) . e − ( a/b ) | γ | . Therefore, (ED) holds. When Γ is hyperb olic and µ has ﬁnite supp ort, A. Ancona [1] pro v ed that the Martin b oundary is homeomorphic to the visual b oundary ∂ X . The k ey p oin t in his pro of is the follo wing estimate (see [49, Thm. 27.12] a nd Theorem 1 .7). Theorem 3.7. ( A . Ancona) L et Γ b e a non-elemen tary hyp erb olic gr oup, X a lo c al ly ﬁnite Cayley gr aph endowe d with a ge o desic metric d s o that Γ acts c anonic al ly by isom etries, and let µ b e a ﬁnitely supp orte d symmetric pr ob ability me a s ur e the supp ort of w hich gener ates Γ . F or any r ≥ 0 , ther e is a c onstant C ( r ) ≥ 1 such that F ( x, v ) F ( v , y ) ≤ F ( x, y ) ≤ C ( r ) F ( x, v ) F ( v , y ) whenever x, y ∈ X and v is at distanc e at most r fr om a ge o desic se gmen t b etwe en x and y . 20 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU This implies together with Lemma 3.6 that when µ is ﬁnitely supp orted, b o th conditions (ED) and (QR) hold. Therefore, Prop o sition 3.5 implie s that d G ∈ D (Γ). W e ha ve just established the ﬁrst statemen t of Corollar y 1.2. 3.4. Martin kerne l vs Busemann function: end of the pro of of Theorem 1.1. W e assume that X = Γ equipp ed with the Green metric d G b elongs to D (Γ ) throughout this paragraph. Notation. When we consider notions with resp ect to d G , w e will add the expo nent G to distinguish them from the same no t io ns in the initial metric d . Th us Buse mann functions for d G will b e written β G a . The visual metric on ∂ X see n from w for the original metric will b e denote b y d ε , and b y d G ε for the one coming fro m d G . Balls at inﬁnit y will be denoted b y B ε and B G ε . Let us recall that the Martin k ernel is deﬁned b y K ( x, y ) = F ( x, y ) F ( w , y ) = exp { d G ( w , y ) − d G ( x, y ) } . By deﬁnition of the Martin b oundary ∂ M X , the k ernel K ( x, y ) contin uously extends to a µ -harmonic p ositive function K a ( · ) when y tends to a p oin t a ∈ ∂ M X . W e recall t hat, b y Theorem 1 .7, w e ma y - a nd will - iden tify ∂ M X with the visual b oundary ∂ X . As w e already men tioned Γ a cts on ∂ M X , so on it s harmonic measure and w e hav e γ ∗ ν = ν γ − 1 . Besides , se e e.g. G . Hunt [24] or W. W o ess [49, Th. 24 .1 0] for what follo ws, ν and ν γ are absolutely con tinuous and their Radon-Nikodym deriv a tiv es satisfy dν γ dν ( a ) = K a ( γ ( w )) . W e already computed the Busemann function in the metric d G in part 3.2: β G a ( w , x ) = log K a ( x ). Th us w e ha v e pro v ed that dγ ∗ ν dν ( a ) = exp β G a ( w , γ − 1 w ) . It follows at once that ν is a quasiconformal measure on ( ∂ X , d G ε ) of dimension 1 /ε . Actually , ν is ev en a c onfo rmal me as ur e since w e ha v e a genuine equalit y ab ov e. Therefore ν b elongs to the P atterson-Sulliv an class asso ciated with the metric d G . According to Theorem 2.3, it is in particular comparable to the Hausdorﬀ measure for t he corresp onding visual metric. This ends b oth the pro ofs of Theorem 1.1 and of Corollary 1.2. W e note that, comparing the statemen ts in Theorem 1 .1 (ii) a nd Theorem 2.3, we rec o v er the equalit y v G = 1 a lr eady noticed in [7] for random w a lks on non-amenable g r o ups. See also [8]. 3.5. C onsequences. W e now dra w consequences of the hyperb o licit y of the Green metric. W e refer to the app endices for pr o p erties o f quasiruled spac es. 3.5.1. Deviation ine qualities. W e study the latera l deviation of the p osition of the random w alk with r esp ect to t he quasiruler [ w , Z ∞ ( w )) where, fo r an y x ∈ X and a ∈ ∂ X , we c hose an arbitrar y quasiruler [ x, a ) from x to a in a measurable w ay . HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 21 Prop osition 3.8. Assume that Γ is a non-elemen tary hyp erb olic gr oup, ( X, d ) ∈ D (Γ) , and µ is a symmetric law so that the asso ciate d Gr e en metric b elong s to D (Γ) . The fo l lowing holds (i) Ther e is a p ositive c ons tant b so that, for any D ≥ 0 and n ≥ 0 , P [ d ( Z n ( w ) , [ w , Z ∞ ( w ))) ≥ D ] . e − bD . (ii) Ther e is a c onstant τ 0 such that for any p ositive inte gers m, n, k , E [( Z m ( w ) | Z m + n + k ( w )) Z m + n ( w ) ] ≤ τ 0 . Proo f. Pro of of (i). Observ e that P [ d ( Z n ( w ) , [ w , Z ∞ ( w ))) ≥ D ] = X z ∈ X P [ d ( Z n ( w ) , [ w , Z ∞ ( w ))) ≥ D , Z n ( w ) = z ] = X z ∈ X P [ d ( z , [ w , Z − 1 n Z ∞ ( z ))) ≥ D , Z n ( w ) = z ] = X z ∈ X P [ d ( z , [ w , Z − 1 n Z ∞ ( z ))) ≥ D ] P [ Z n ( w ) = z ] = X z ∈ X P [ d ( z , [ w , Z ∞ ( z ))) ≥ D ] P [ Z n ( w ) = z ] The second equalit y holds b ecause γ w = z implies that γ − 1 Z ∞ ( z ) = Z ∞ ( w ). The t hir d equalit y comes fro m the indep endence of Z n = X 1 X 2 · · · X n and Z − 1 n Z ∞ = X n +1 X n +2 · · · . The last equalit y uses t he fact that Z − 1 n Z ∞ and Z ∞ ha v e the same law . On the eve n t { d ( z , [ w , Z ∞ ( z ))) ≥ D } , we ha v e in par ticular d ( w , z ) ≥ D and we can pic k x ∈ [ w , z ) suc h that d ( z , x ) = D + O (1). Then, b ecause the triangle ( w , z , Z ∞ ( z )) is thin and since d ( z , [ w , Z ∞ ( z ))) ≥ D , we m ust hav e Z ∞ ( z ) ∈ ℧ z ( x, R ). As usual R is a constant t ha t do es not dep end on z , D or Z ∞ ( z ). W e now apply the lemma of the shadow Lemma 2.4 to the G r een metric to ded uce that P [ d ( z , [ w , Z ∞ ( z ))) ≥ D ] ≤ P z [ Z ∞ ( z ) ∈ ℧ z ( x, R )] = ν z ( ℧ z ( x, R )) . e − d G ( z ,x ) . Finally , using the quasi-isometry b etw een d a nd d G , it follo ws that P [ d ( Z n ( w ) , [ w , Z ∞ ( w ))) ≥ D ] . e − bD . Pro of of (ii). Using the indep endence of the incremen ts o f the w alk, one ma y ﬁrst a ssume tha t m = 0. Let us choose Y n ( w ) ∈ [ w , Z ∞ ( w )) suc h that d ( w , Y n ( w )) is as close from ( Z n ( w ) | Z ∞ ( w )) as p ossible. Since t he space ( X , d ) is quasiruled, it follo ws that d ( w , Y n ( w )) = ( Z n ( w ) | Z ∞ ( w )) + O (1). (W e only use Landau’s notation O (1) for estimates that are uniform with resp ect to the tra jectory of ( Z n ). Th us the line just ab ov e means that there exists a determinis tic constan t C suc h that | d ( w , Y n ( w )) − ( Z n ( w ) | Z ∞ ( w )) | ≤ C . The same con v ention applies to the rest of the pro o f .) Let us deﬁne A 0 = { d ( w , Y n ( w )) ≤ d ( w , Y n + k ( w )) } 22 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU and, f or j ≥ 1, A j = { j − 1 < d ( w , Y n ( w )) − d ( w , Y n + k ( w )) ≤ j } . Appro ximating { w , Z n ( w ) , Z n + k ( w ) , Z ∞ ( w ) } by a tree, it f ollo ws that, on the ev en t A 0 , ( w | Z n + k ( w )) Z n ( w ) ≤ d ( Z n ( w ) , [ w , Z ∞ ( w ))) + O (1) and that, on the ev en t A j , ( w | Z n + k ( w )) Z n ( w ) ≤ d ( Z n ( w ) , [ w , Z ∞ ( w ))) + j + O (1) . Therefore E [( w | Z n + k ( w )) Z n ( w ) ] ≤ E [ d ( Z n ( w ) , [ w , Z ∞ ( w )))] + X j ≥ 1 j P ( A j ) + O (1) . If d ( w , Y n ( w )) − d ( w , Y n + k ( w )) ≥ j then d ( Z n + k ( w ) , [ Z n ( w ) , Z ∞ ( w ))) ≥ j so that P ( A j +1 ) ≤ P [ d ( Z n + k ( w ) , [ Z n ( w ) , Z ∞ ( w ))) ≥ j ] . Using (i) for the random walk starting at Z n ( w ), w e get X j ≥ 1 j P ( A j ) . 1 . On t he other hand, E [ d ( Z n ( w ) , [ w , Z ∞ ( w )))] = Z ∞ 0 P [ d ( Z n ( w ) , [ w , Z ∞ ( w ))) ≥ D ] dD . Z ∞ 0 e − bD dD = 1 / b . The prop o sition f o llo ws. W e now improv e the estimate (2) in Theorem 3.1 when d G ∈ D (Γ). Corollary 3.9. L et Γ b e a non-elem entary hyp erb olic gr oup, ( X , d ) ∈ D (Γ) and µ a symmetric law such that d G ∈ D (Γ) , then we have (6) lim sup d ( Z n ( w ) , [ w , Z ∞ ( w ))) log n < ∞ P a.s. Proo f. It follo ws from Proposition 3.8 that w e ma y ﬁnd a constant κ > 0 so that P [ d ( Z n ( w ) , [ w , Z ∞ ( w ))) ≥ κ log n ] ≤ 1 n 2 . Therefore, the Borel-Can telli lemma implies that lim sup d ( Z n ( w ) , [ w , Z ∞ ( w ))) log n < ∞ P a.s. and the corollary follows. Remark. After a ﬁrst v ersion of this pap er w as publicised, M. Bj o rklund also used the h yperb olicit y of the Green metric to prov e a Central Limit Theorem f o r d G ( w , Z n ( w )), see [6]. HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 23 3.5.2. Esc ap e of the r andom walk fr om b al ls. W e assume here that µ is a symmetric and ﬁnitely supp orted probability measure on a non-elemen tar y h yperb olic group Γ and that the support of µ g enerates Γ. W e w an t to compare the har mo nic measure with the uniform measure on the spheres for the Green metric. W e deﬁne the (exterior) sphere of the ba ll B G ( w , R ) by ∂ B G ( w , R ) def. = { x ∈ X : x 6∈ B G ( w , R ) and ∃ γ ∈ S upp ( µ ) s.t. γ − 1 ( x ) ∈ B G ( w , R ) } . The harmonic measure ν R on ∂ B G ( w , R ) is t he la w of the ﬁrst p oin t visited outside B G ( w , R ) . As the v olume o f the sphere ∂ B G ( w , R ) equals e R up to a multiplicativ e constant (see [7]), w e need to compare ν R ( · ) with e − R . In other w ords, w e ha v e to bound the ratio b et w een the measure ν R ( · ) and the hitting probability F ( w , · ). O bserv e that, in principle, there could b e p oin ts on the sphere that are visited by the w alk a long t ime after it left the ball. W e shall see that this scenario can only t ak e pla ce on a ﬁnite sc ale. In the follow ing w e only consider quasigeo desics fo r ( X , d ) a nd ( X , d G ) that are geo desics for a giv en w ord metric d w ∈ D (Γ). Prop osition 3.10. Ther e exis t p ositive c o n stants C 1 < 1 and C 2 such that for any p ositive r e a l R , the harmonic me asur e ν R on the spher e ∂ B G ( w , R ) satisﬁes ∀ x ∈ ∂ B G ( w , R ) , ∃ y ∈ B G ( x, C 1 ) ∩ ∂ B G ( w , R ) s.t. C 2 e − R ≤ ν R ( y ) ≤ e − R . Proo f. The upp er b ound (v alid for an y x ∈ ∂ B G ( w , R ) ) ob viously follow s from the deﬁnition of the Green metric: if y 6∈ B G ( w , R ) , t hen ν R ( y ) ≤ F ( w , y ) = exp ( − d G ( w , y )) ≤ e − R . F or the lo w er b ound, w e consider a quasigeo desic from w to x and denote b y y the ﬁrst p oin t of ∂ B G ( w , R ) along that path. Since µ has ﬁnite supp o rt, d G ( w , x ) and d G ( w , y ) only diﬀer by an a dditive constant. The quasiruler prop ert y then implies that y is at a b ounded distance f rom x . Let E = E ( R ) denote the set of p oints z ∈ ∂ B G ( w , R ) suc h that there is a quasigeodesic reac hing z from w en tirely con ta ined in B G ( w , R ) ( except for the la st step to w ard z ). Let z ∈ E . Since y and z b elong to ∂ B G ( w , R ) , t hen d G ( w , z ) a nd d G ( w , y ) only diﬀer b y an additive constan t and w e hav e (7) d G ( y , z ) ≥ d G ( y , z ) + ( d G ( w , y ) − d G ( w , z ) − C ) = 2 ( w | z ) y − C Let k 0 b e an in teger a nd deﬁne E 0 def. = { z ∈ E : ( w | z ) y ≤ k 0 } and for all inte ger k ≥ k 0 , E k def. = { z ∈ E : k < ( w | z ) y ≤ k + 1 } . W e denote b y τ R the ﬁrst hitting time of ∂ B G ( w , R ) by the random w a lk a nd by τ y the ﬁrst hitting time of y . Then F ( w , y ) = P [ τ y < ∞ , Z τ R ( w ) ∈ E 0 ] + ∞ X k = k 0 X z ∈E k P [ τ y < ∞ , Z τ R ( w ) = z ] A t this p oin t, we need to use the Stro ng Mark o v pr o p ert y to sa y that once w e kno w that Z τ R ( w ) = z and z 6 = y , the hitt ing time of y m ust o ccur aft er τ R . Then, the ﬁniteness of τ y 24 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU dep ends only on the position z disregarding the b eha vior of the random w alk up to time τ R . Namely , P [ τ y < ∞ , Z τ R ( w ) = z ] = P z [ τ y < ∞ ] P [ Z τ R ( w ) = z ] . Using (7) , the deﬁnition of ( E k ) a nd the inequalit y P [ Z τ R ( w ) = z ] ≤ P [ τ z < ∞ ] ≤ e − R , we get that (8) F ( w , y ) ≤ P [ Z τ R ( w ) ∈ E 0 ] + C ∞ X k = k 0 e − 2 k e − R # E k . W e need an upp er b ound on # E k . T ak e z ∈ E k , and let y R − k b e the p oin t at distance R − k from w a lo ng the quasigeodesic [ w , y ]. As the triangle ( w , z , y ) is thin, the center of t he asso ciated approximate tree is at a b ounded distance fro m the p oin t y R − k . Then, since for an y z in E k , ( w | y ) z − k is b ounded by a constan t, the set E k is therefore included in the ba ll B G ( y R − k , k + C ) f o r some constan t C . Th us # E k . e k and (9) C ∞ X k = k 0 e − 2 k e − R # E k ≤ C ( k 0 ) e − R with C ( k 0 ) tending to 0 when k 0 tends to inﬁnit y . As µ is ﬁnitely supp o rted, ∂ B G ( w , R ) is at a b ounded distance from B G ( w , R ) . So y ∈ B G ( w , R + C ( µ )) and F ( w , y ) ≥ e − C ( µ ) e − R . No w c ho ose k 0 so that C ( k 0 ) < (1 / 2) e − C ( µ ) and tak e R > k 0 . Then (8) and (9) giv e (10) P [ Z τ R ( w ) ∈ E 0 ] ≥ 1 2 e − C ( µ ) e − R . W e conclude that ν R ( E 0 ) & e − R . T ak e y ′ ∈ E 0 so that ( w | y ′ ) y ≤ k 0 . By the deﬁnition of the set E a nd b y the thinnes s of the triangle ( w, y , y ′ ), there exists a path joining y and y ′ within B G ( w , R ) of length at most c ( k 0 ), a constan t dep ending only on k 0 and δ . Therefore, there exists a constan t c ′ ( k 0 , µ ) suc h that ν R ( y ) ≥ ν R ( y ′ ) c ′ ( k 0 , µ ) . Finally , as # E 0 is b ounded abov e b y a constan t, (10) gives ν R ( y ) & X y ′ ∈E 0 ν R ( y ′ ) = ν R ( E 0 ) & ε − R . Remark. Prop osition 3.10 sa ys that the harmonic measure on spheres is w ell spread out and that the harmonic measure of a b ounded domain o f the sphere of radius R if e − R up to a m ultiplicativ e constan t. Appro ximating the balls of ∂ X b y shado ws, w e get that ν is Alfhors- regular of dimension 1 /ε , hence quasiconformal. The refore, w e get an alt ernativ e pro of of the second statement of The orem 1.1 when µ has ﬁnite supp o rt. 3.5.3. The do ubling c ondition for the harmonic me asur e. Let us recall that a measure m is said to b e doubling if there exists a constant C > 0 such that, for any ball B of radius at most the diameter of the space then m (2 B ) ≤ C m ( B ). Prop osition 3.11. L et Γ b e a non-elementary hyp erb olic gr oup, ( X, d ) ∈ D (Γ) and let µ b e a symmetric la w such that d G ∈ D (Γ) . The harmonic me asur e is doublin g with r esp e ct to the visual me asur e d ε on ∂ X . HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 25 Proo f. The mo dern formulation of Efremo vic h and Tich onirov a’s theorem (cf. T heorem 6.5 in [10] and references therein) states that quasi-isometries betw een h yperb olic prop er geo desic spaces Φ : X → Y extend as quasisymmetric maps φ : ∂ X → ∂ Y b etw een their visual b oundaries i.e., there is an increasing homeomorphism η : R + → R + suc h that | φ ( a ) − φ ( b ) | ≤ η ( t ) | φ ( a ) − φ ( c ) | whenev er | a − b | ≤ t | a − c | . Since d G ∈ D (Γ ) , the spaces in v olv ed are visual. Th us, the statemen t remains true since w e ma y still appro ximate prop erly the space by trees, cf. Appendix B. Since ( X, d ) and ( X, d G ) a re quasi-isometric, the b oundaries are thus quasisymmetric with resp ect to d ε and d G ε . F urthermore, ν is doubling with respect to d G ε since it is Ahlfors-regular, and this prop ert y is preserv ed under quasisym metry . Basic prop erties on quasisymmetric maps include [23]. More information on b oundaries of h yperb olic groups, and the relationships b et w een h yp erb olic geometry and conformal geometry can b e found in [11, 31]. 4. Dimension of the harmonic measure on the boundar y of a hyp erbolic metric sp ace Theorem 1 .3 will follo w from Prop osition 4.1 and Prop osition 4.2 . W e recall the deﬁnition of the rates of esc ap e ℓ and ℓ G of the random walk with respect to d or d G resp ectiv ely . ℓ def. = lim n d ( w , Z n ( w )) n and ℓ G def. = lim n d G ( w , Z n ( w )) n . W e will ﬁrst pro ve Prop osition 4.1. L e t Γ b e a non- elementary hyp erb olic gr oup and let ( X , d ) ∈ D (Γ) . L et µ b e a symmetric pr ob ability me asur e on Γ the supp ort of which gener ates Γ such that d G ∈ D (Γ) and with ﬁnite ﬁrst moment X γ ∈ Γ d G ( w , γ ( w )) µ ( γ ) < ∞ . L et ν b e the harmonic me asur e se en fr o m w on ∂ X . F or ν -a.e. a ∈ ∂ X , lim r → 0 log ν ( B ε ( a, r )) log r = ℓ G εℓ , wher e B ε denotes the b al l on ∂ X for the visual metric d ε . Remark. Recall from [8] that µ having ﬁnite ﬁrst momen t with resp ect to the Green metric is a conseq uence of µ ha ving ﬁnite en tr o p y . Proo f. It is con v enien t t o in tro duce an auxiliary w or d metric d w whic h is of course geo desic. W e may then consider the visual quasiruling structure G induced b y the d w -geo desics f or b oth metrics d and d G via the iden tit y map, cf. the a pp endix. W e combine Prop ositions 2 .1 and B.5 to get that , for a ﬁxed but larg e enough R , for any a ∈ ∂ X and x ∈ [ w , a ) ⊂ G B ε ( a, (1 /C ) e − εd ( w ,x ) ) ⊂ ℧ G ( x, R ) ⊂ B ε ( a, C e − εd ( w ,x ) ) 26 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU and B G ε ( a, (1 /C ) e − εd G ( w, x ) ) ⊂ ℧ G ( x, R ) ⊂ B G ε ( a, C e − εd G ( w, x ) ) for some p o sitiv e constant C . W e recall that the shado ws ℧ G ( x, R ) a re deﬁned using geo desics for the w o rd metric d w . The do ubling pro p ert y of ν with resp ect to the visual metric d ε implies tha t (11) ν ( B ε ( a, C e − εd ( w ,x ) )) ≍ ν ( ℧ G ( x, R )) for any x ∈ [ w , a ). Let η > 0; by deﬁnition of the drift, there is a set of full measure with r esp ect to the la w of the tra j ectories of the random walk, in whic h for a n y sequence ( Z n ( w )) and for n large enough, we ha v e | d ( w , Z n ( w )) − ℓn | ≤ η n and | d G ( w , Z n ( w )) − ℓ G n | ≤ η n . F rom Theorem 3.1 applied to the metric s d and d G , w e get that, for n large enoug h, d ( Z n ( w ) , π n ( Z ∞ ( w ))) ≤ η n and d G ( Z n ( w ) , π n ( Z ∞ ( w ))) ≤ η n . W e conclude that (12)  | d ( w , π n ( Z ∞ ( w ))) − ℓn | ≤ 2 η n | d G ( w , π n ( Z ∞ ( w ))) − ℓ G n | ≤ 2 η n Set r n = e − εd ( w ,π n ( Z ∞ ( w ))) . Therefore, us ing (11) with a = Z ∞ ( w ) and x = π n ( Z ∞ ( w )), w e get ν ( B ε ( Z ∞ ( w ) , r n )) ≍ ν ( ℧ G ( π n ( Z ∞ ( w )) , R )) ≍ e − d G ( w, π n ( Z ∞ ( w ))) where the r ig h t-hand pa r t comes from the fact t ha t ν is a quasiconformal measure of dimension 1 /ε for the Green visual metric a nd the lemma of the shado w (Lemma 2.4). Hence w e deduce from (12) that, if n is large enough, then (13)     log ν ( B ε ( Z ∞ ( w ) , r n )) log r n − ℓ G εℓ     . η . Since the measure ν is doubling (Prop osition 3.11) , ν is a lso α -homogeneous for some α > 0, (cf. [23, Chap. 13]) i.e., there is a constan t C > 0 suc h that , if 0 < r < R < diam ∂ X and a ∈ ∂ X , then ν ( B ε ( a, R )) ν ( B ε ( a, r )) ≤ C  R r  α . F rom     log e − εnℓ r n     ≤ 2 nεη it follows that     log ν ( B ε ( Z ∞ ( w ) , e − εnℓ )) ν ( B ε ( Z ∞ ( w ) , r n ))     ≤ 2 nαεη + O (1) . Therefore lim sup n     log ν ( B ε ( Z ∞ ( w ) , e − εnℓ )) log e − εnℓ − log ν ( B ε ( Z ∞ ( w ) , r n )) log r n     . η . Since η > 0 is arbitr ary , it follows from (13) that lim r → 0 log ν ( B ε ( Z ∞ ( w ) , r )) log r = lim n →∞ log ν ( B ε ( Z ∞ ( w ) , e − εnℓ )) log e − εnℓ = lim n →∞ log ν ( B ε ( Z ∞ ( w ) , r n )) log r n = ℓ G εℓ . HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 27 In other w ords, f o r ν almost ev ery a ∈ ∂ X , lim r → 0 log ν ( B ε ( a, r )) log r = ℓ G εℓ . It remains to prov e that ν has dimens ion ℓ G /εℓ . This is standard. Hausdorﬀ measures. Let s, t ≥ 0, w e set H t s ( X ) def. = inf n X r s i , B i = B ( x i , r i ) , X ⊂ ( ∪ B i ) , r i ≤ t o , where we consider cov ers b y ba lls. The s - dimensional measure is then H s ( X ) def. = lim t → 0 H t s ( X ) = sup t> 0 H t s ( X ) . The Hausdorﬀ dimension dim H X of X is the n umber s ∈ [0 , ∞ ] suc h that, for s ′ < s , H s ′ ( X ) = ∞ holds and for all s ′ > s , H s ′ ( X ) = 0. The Hausdorﬀ dimension dim ν of a measure ν is the inﬁm um of the Hausdorﬀ dimensions o v er all sets of full measure. Replacing co v ers b y balls by co v ers b y any kind of sets in the deﬁnition of H t s ( X ) and replacing ra dii b y diameters w o uld not c hange the v alue of dim ν . F or more prop erties, one can consult [42]. Prop osition 4.2. L et X b e a pr op er metric sp a c e and ν a Bor el r e gular pr ob ability me asur e on X . If, for ν -alm ost every x ∈ X , lim r → 0 log ν ( B ( x, r ) ) log r = α then dim ν = α . W e recall the pro of for the con v enience of the reader. W e will use the follo wing cov ering lemma. Lemma 4.3. L et X b e a pr op er metric sp ac e and B a family of b al ls in X with uniform l y b ounde d r a d ii. Th e n ther e is a subfamily B ′ ⊂ B of p airwis e disjo i n t b al ls such that ∪ B B ⊂ ∪ B ′ (5 B ) . F or a pro of of the lemma, see Theorem 2.1 in [42]. Proo f of Prop. 4.2. Let s > α , and c ho ose η > 0 small enough so that β := s − α − η > 0. F or ν - almost ev ery x , a radius r x > 0 exists so that     log ν ( B ( x, r )) log r − α     ≤ η , for r ∈ (0 , r x ]. Let us denote by Y = { x ∈ X : r x < ∞} , whic h is of full measure. Let us ﬁx t ∈ (0 , 1). F or an y x ∈ Y , w e ch o ose ρ x = min { r x , t } . W e apply Lemma 4.3 to { B ( x, ρ x ) } and obtain a 28 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU subfamily B t . It follo ws that Y is co v ered b y 5 B t and H 5 t s ( Y ) ≤ X B t (5 ρ x ) s ≤ 5 s t β X B t ρ α + η x . t β X B t ν ( B ( x, ρ x )) . t β ν ( ∪ B t B ( x, ρ x )) . t β whic h tends to 0 with t . Therefore H s ( Y ) = 0 a nd so dim H Y ≤ s for all s > α . Whence dim ν ≤ α . Con v ersely , let Y b e a se t of full measure. There is a subset Z ⊂ Y suc h that ν ( Z ) ≥ 1 / 2 and suc h tha t the con v ergence of log ν ( B ( x, r )) / log r to α is uniform on Z (Egorov theorem). Fix s < α and let us consider η > 0 small enough so that γ = α − η − s > 0 . There exists 0 < r 0 ≤ 5 suc h that, for any r ∈ (0 , r 0 ) a nd any x ∈ Z ,     log ν ( B ( x, r )) log r − α     ≤ η . Let B be a cov er o f Z b y balls of radius ρ x smaller than t ≤ r 0 / 5. Pic k a subfamily B t = { B ( x, ρ x ) } usin g Lemma 4.3. Then 5 B t co v ers Z and 1 / 2 ≤ X B t ν (5 B ) ≤ 5 α − η X B t ρ α − η x . t γ X B t ρ s x . This prov es that H t s ( Z ) & t − γ so that dim H Y ≥ dim H Z ≥ α . 5. Harmonic measure of maximal dimension This section is devoted to the pro of of Theorem 1.5 and its corollary . 5.1. T he funda men tal equalit y. W e assume that d ∈ D (Γ), µ is a probabilit y measure with exp onential moment suc h that d G ∈ D (Γ ) . Thu s there exists λ > 0 suc h that E def. = E  e λd ( w,Z 1 ( w ))  < ∞ . The main issue in the pro of of Theorem 1.5 is the follow ing implication which w e prov e ﬁrst: Prop osition 5.1. Under the hyp otheses of The or em 1.5, if h = ℓv , then ρ and ν ar e e quivalent. Let R b e t he constant coming f rom the lemma of the shado w (Lemma 2.4) and write ℧ ( x ) for ℧ ( x, R ) . Let us no w deﬁne ϕ n = ρ ( ℧ ( Z n ( w ))) ν ( ℧ ( Z n ( w ))) and φ n = log ϕ n . Since µ n is the la w of Z n , o bserv e that, if β ∈ (0 , 1 ], t hen E [ ϕ β n ] = X γ ∈ Γ µ n ( γ )  ρ ( ℧ ( γ ( w ))) ν ( ℧ ( γ ( w ) ) )  β and E [ φ n ] = X γ ∈ Γ µ n ( γ ) log  ρ ( ℧ ( γ ( w ))) ν ( ℧ ( γ ( w ) ) )  . W e start with tw o lemmata. HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 29 Lemma 5.2. Ther e ar e ﬁ nite c onstants C 1 ≥ 1 and β ∈ (0 , 1] such that, for al l N ≥ 1 , 1 N X 1 ≤ n ≤ N E [ ϕ β n ] ≤ C 1 . When µ is ﬁnitely suppor t ed, one can ch o ose β = 1 in the lemma. Proo f. Let N ≥ 1 and 1 ≤ n ≤ N b e chosen. W e will ﬁrst pro v e that there are some κ and β indep enden t from N and n suc h that (14) R κ def. = X γ , d ( w, γ ( w )) ≥ κN  ρ ( ℧ ( γ ( w ))) ν ( ℧ ( γ ( w ) ) )  β µ n ( γ ) . 1 . W e hav e already seen t ha t the log arithmic v olume gro wth rate for the Green metric is 1. Then, f r om the lemma of the shado w (Lemma 2.4) applied to b oth metrics, w e get (15) ν ( ℧ ( γ ( w ) ) ) ≍ e − d G ( w, γ ( w )) = F ( w , γ ( w )) ≍ G ( w , γ ( w )) = X k µ k ( γ ) and (16) ρ ( ℧ ( γ ( w ))) ≍ e − vd ( w ,γ ( w )) . On the o ther ha nd, since d G is quasi-isometric to d , it follow s that there is a constan t c > 0 suc h that ρ ( ℧ ( γ ( w ))) ν ( ℧ ( γ ( w ) ) ) . e cd ( w , γ ( w )) . Hence R κ . X k ≥ κN e cβ k X k ≤ d ( w, γ ( w )) 1 suc h that the set A def. = { (1 /C ) ≤ ˜ J ≤ C } has p ositiv e ˜ ν -measure. Since ˜ ν is ergodic, for ˜ ν -almost ev ery ( a, b ) ∈ ∂ 2 X , there exists γ ∈ Γ suc h that γ ( a, b ) ∈ A . It follo ws from the inv aria nce of ˜ ν a nd the quasi-in v ariance of ˜ ρ that ˜ J ( a, b ) ≍ ˜ J ( γ ( a ) , γ ( b )) . This prov es the claim. Therefore, for ˜ ν -almost ev ery ( a, b ), J ( a ) J ( b ) ≍ exp 2 v ( a | b ) exp 2( a | b ) G . Let us assume tha t log J is un b ounded in a neigh b orho o d U of a point a ∈ ∂ X . W e may ﬁnd a p o int b ∈ ∂ X with J ( b ) ﬁnite and non-zero, and f a r enough from U so that exp 2 v ( c | b ) exp 2( c | b ) G ≍ 1 for any c ∈ U . This pro v es that lo g J had to b e b ounded in U : a con tradiction. 5.3. Geometr ic c haracterisation of t he fundamen t al inequalit y. W e may no w turn to the pro of of Theorem 1.5. Proo f of Theorem 1.5. W e ﬁrst prov e that (i), ( ii) and (iii) are equiv alen t. Then w e prov e that (iii) implies (iv), (iv) implies (v) whic h implies (iii). • F rom Prop osition 5.1, w e deduce that (i) implies (ii). Prop osition 5.4 say s t hat (ii) implies (iii). F urthermore, if ν and ρ are equiv alen t, then they hav e the same Hausdorﬀ dimension. So, from Corollary 1.4 and Theorem 2.3, w e get that h ℓε = dim ν = dim ρ = v ε , and t hus h = ℓv . • T o prov e that (iii) implies (iv), we apply the lemma of the shadow (Lemma 2.4): it follo ws that, for any γ ∈ Γ, e − vd ( w ,γ ( w )) ≍ ρ ( ℧ ( γ ( w ))) ≍ ν ( ℧ ( γ ( w ))) ≍ e − d G ( w, γ ( w )) whence the existence o f a constan t C suc h that | v d ( w , γ ( w )) − d G ( w , γ ( w )) | ≤ C . 34 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU Since Γ acts transitiv ely b y isometries for b oth metrics, it follows that ( X , v d ) and ( X , d G ) a re (1 , C )- quasi-isometric. • Assuming (iv), it f o llo ws t hat Busemann functions coincide up to the multiplicativ e factor v . Therefore, the Radon-Nik o dym deriv a t ive of γ ∗ ν with respect to ν at a p oin t a ∈ ∂ X is prop or t ional to exp( − v β a ( w , γ − 1 ( w ))) a.e. Therefore, ν is a quasiconformal measure f o r ( ∂ X , d ε ). This is (v). • F or the last implicatio n, (v) implies (iii) , one can use the uniqueness statemen t in Theorem 2.3 to get t ha t ρ and ν a re equiv alen t and hav e b ounded densit y . This prov es (iii). 5.4. Simultan eous random w alks. W e now turn to the pro of of Corollary 1 .6. Proo f of Corollar y 1. 6. Let us consider the Green metric d G asso ciated with µ a nd denote b y b ℓ the drift of ( b Z n ) in the metric space (Γ , d G ). Theorem 1 .1 implies that d G ∈ D (Γ). Assumption (i) tr a nslates into b h = b ℓ . Since v G = 1, this means that b ν has maximal dimension in the b oundary of ( Γ , d G ) endow ed with a visual metric. Therefore Theorem 1.5 implies the equiv a lence betw een (i) a nd (iii). Exc hang ing the roles of µ and b µ giv es the equiv alence b et w een (ii) and (iii). If b d G denotes the Green metric fo r b µ , then (iv) means that d G and b d G are (1 , C )- quasi- isometric, wh ic h is equiv alen t to (iii) b y Theorem 1.5. 5.5. F uch sian groups with cusps. The next prop osition is the ke y to the pro of of Theorem 1.11. Let us ﬁrst in tro duce its setting. Let X b e a prop er quasiruled hyperb olic space a nd let Γ b e a hy p erb olic subgroup of isome- tries tha t acts prop erly discontin uously on X . Consider a symmetric probability measure µ o n Γ with ﬁnite supp ort and whose supp ort generates the gr oup Γ. Let ν b e the corresp onding harmonic measure on ∂ Γ, the vis ual boundary of Γ. Let Γ( w ) b e the orbit of some p oin t w ∈ X . As for Theorem 3 .1, Theorem 7.3 in [30] a nd § 7.4 therein imply that the seq uence Z n ( w ) almost surely con v erges to some p oint Z ∞ ( w ) in ∂ X , the visual b oundary of X . Let ν 1 b e the la w of Z ∞ ( w ). Although the tw o spaces ∂ X and ∂ Γ migh t b e to p ologically diﬀeren t, the t w o measured spaces ( ∂ X, ν 1 ) and ( ∂ Γ , ν ) are isomorphic as Γ- spaces i.e., there exists a measured spaces isomorphism Φ from ( ∂ Γ , ν ) to ( ∂ X , ν 1 ) that conjugates the action of Γ on b oth spaces. Indeed bo th spaces are mo dels for the P oisson b oundary of the random w alk. This is pro ve d in [30] Theorem 7.7 and Remark 3 following it for ( ∂ X , ν 1 ) and it is a general fact for the Martin b oundary ( ∂ Γ , ν ). Prop osition 5.5. L et X b e a pr op er quasirule d hyp erb olic sp ac e end o we d with a ge ometric gr oup action. L et ρ b e the c orr esp ond i n g Patterson-Sul li v an me a s ur e. L et Γ b e a hyp erb olic sub gr o up of isom etries that acts pr op erly di s c ontinuously on X an d Γ( w ) b e an orbit of Γ in X . L et µ b e a symmetric pr ob ability me asur e on Γ with ﬁnite supp ort and whose supp ort gen e r ates the gr oup Γ . L et ν 1 b e the limit law of the tr aj e ctories of the r andom wal k on ∂ X . I f ρ and ν 1 ar e e quivale n t then Γ and Γ( w ) ar e quasi-is o metric. Proo f. One che c ks as in Prop osition 5.4 that , once ρ and ν 1 are equiv alen t, then their densit y is almost surely b ounded. HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 35 W e recall the following c hange of v ariables form ula: dγ ∗ ν dν ( a ) = K a ( γ − 1 ) , for ν almost an y p oint a ∈ ∂ Γ a nd where K a is the Martin k ernel. Because of the isomorphism Φ, we also hav e dγ ∗ ν 1 dν 1 ( ξ ) = K Φ − 1 ( ξ ) ( γ − 1 ) , for ν 1 almost a n y p oint ξ ∈ ∂ X . On t he other hand, ρ b eing a quasiconformal measure, it satisﬁes dγ ∗ ρ dρ ( ξ ) ≍ e vβ ξ ( w, γ − 1 ( w )) , where β ξ is the Busemann function in X . Since the densit y of ν 1 with resp ect to ρ is b ounded and b ounded a w ay fro m 0, w e therefore ha v e (20) K Φ − 1 ( ξ ) ( γ − 1 ) ≍ e vβ ξ ( w, γ − 1 ( w )) , for ρ almost an y ξ . W e now use Lemma B.6. First o bserv e that sup ξ ∈ ∂ X β ξ ( x, y ) can b e replaced b y an essen tial sup with resp ect to ρ since ρ , b eing quasiconformal, c harges any non empt y ball and since ξ → β ξ ( x, y ) is locally almost constant. So w e get from Lemma B.6 that | d ( x, y ) − v ess sup ξ ∈ ∂ X β ξ ( x, y ) | is b ounded. By a similar argumen t, applying Lemma B.6 to the Green metric on Γ, w e deduce that | d G ( e, γ − 1 ) − ess sup ξ ∈ ∂ Γ log K ξ ( γ − 1 ) | is b ounded. The esse n tial sup is tak en with r esp ect to ν . But (2 0) implies that | ess sup a ∈ ∂ Γ log K a ( γ − 1 ) − v ess sup ξ ∈ ∂ X β ξ ( w , γ − 1 ( w )) | is b ounded and therefore sup γ ∈ Γ | d G ( e, γ − 1 ) − v d ( w , γ − 1 ( w )) | < ∞ . W e conclude that Γ and Γ( w ) are indeed quasi-isometric. Proo f of The ore m 1. 11. W e pro ceed by contradiction and assume that ν 1 is equiv alent to the Lebesgue measure λ on S 1 . First note that w e can restrict our attention to the subgroup generated b y the supp ort of µ . If this subgroup turned out to ha v e inﬁnite cov olume then its b oundary w ould b e a strict subset of S 1 and ν 1 w ould certainly not be equiv alent to the Leb esgue measu re. There fore we ma y , and will, assume that the supp ort of µ generates Γ and that Γ has ﬁnite cov olume and is ﬁnitely generated. W e know from Selb erg’s lemma that G con tains a torsion-free ﬁnite subgroup Γ S of ﬁnite index so that H 2 / Γ S is a compact Riemann surface with ﬁnitely many punctures. T herefore 36 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU Γ S is isomorphic to a free group so that Γ is h yp erb olic and its bo undary is a p erfect, to t a lly disconnected, compact set (a Can tor set). Let Γ( w ) b e an orbit of Γ in H 2 . By the ﬁnite co v olume assumption, the limit set of Γ( w ) is homeomorphic to the circle S 1 . But it follo ws from Prop o sition 5 .5 that Γ is quasi-isometric to Γ( w ). As a conse quence the limit set of Γ( w ) is also homeomorphic to the b oundary of Γ. As S 1 is not a Can tor set, w e get the contradiction we w ere lo oking for. 6. D is cretisa tion of Bro wnian motion W e let M b e the univ ersal cov ering of a Riemannian manifold N of pinc hed negativ e curv a- ture and ﬁnite v olume with dec k transformation group Γ i.e., M / Γ = N . W e let d denote the distance deﬁned b y the Riemannian structure o n M . Note that when N is compact, Γ acts geometrically on M , and since it ha s negative curv at ur e, it follo ws that Γ is h yp erb olic and that M is quasi-isometric to Γ by ˇ Sv arc-Milnor’s lemma (Lemma 2.2). W e conside r the diﬀusion pro cess ( ξ t ) generated b y the Laplace-Beltrami op erator ∆ on M . That is, we let p t b e the fundamental solution of the heat equation ∂ t = ∆. Then there is a probabilit y measure P y on the family Ξ y of con tin uous curv es ξ : R + → M with ξ 0 = y suc h that, for an y Borel sets A 1 , A 2 , ..., A n , a nd any times t 1 < t 2 < . . . < t n , P y ( ξ t 1 ∈ A 1 , . . . , ξ t n ∈ A n ) = Z A 1 Z A 2 . . . Z A n p t 1 ( y , x 1 ) p t 2 − t 1 ( x 1 , x 2 ) . . . p t n − t n − 1 ( x n − 1 , x n ) dx 1 . . . dx n . If µ is a p ositiv e measure on M , w e write P µ = R M P y µ ( dy ), and this deﬁnes a measure on the set of Bro wnian paths Ξ. As fo r random walks , the f ollo wing limit exists almost surely and in L 1 and w e call it the drift of the Brow nian motion: ℓ M def. = lim d ( ξ 0 , ξ t ) t ; it is also known that ℓ M > 0 and tha t ( ξ t ) almost surely con v erges to a p oin t ξ ∞ in ∂ M [45, 44]. The distribution o f ξ ∞ is the harmonic measure. F urthermore, V. Kaima novic h has deﬁned an asymptotic en t r op y h M whic h shares the same prop erties as for random w alks [25]: for any y ∈ M , h M def. = lim − 1 t Z p t ( y , x ) log p t ( y , x ) dx . He also prov ed that the fundamen t al inequalit y h M ≤ ℓ M v remains v alid in t his setting, where v denotes the logarithmic volume growth rate of M . 6.1. T he discretised motion. W. Ballmann and F. Ledrappier hav e reﬁned a metho d of T. Ly ons a nd D. Sulliv an [40], further studied b y A. Ancona [1], V. Kaimanovic h [27], and b y A. Karlsson a nd F. Ledrappier [3 2] whic h replaces the Brownian motion b y a random w alk on Γ [3]. The construction go es as follow s in our sp eciﬁc case. Let π : M → N b e the univ ersal cov ering and let us ﬁx a base p oin t w ∈ M . Fix ε > 0 smaller than the injectivit y r adius o f N at π ( w ), and consider V = B ( π ( w ) , ε ) in N ; for D large enough, the set F = { G V ( π ( w ) , · ) ≥ D } is compact in V , where G V denotes the Green function of the Bro wnian motion killed outside V . There exists a so-called Harnack constan t C < ∞ suc h that, f or any p o sitiv e harmonic function h on V and an y p oints a, b ∈ F , h ( a ) /h ( b ) ≤ C holds. HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 37 Let V = π − 1 ( V ), F = π − 1 ( F ), V x = B ( x, ε ) and F x = F ∩ V x for x ∈ X def. = Γ( w ). If y ∈ F x , w e set χ ( y ) = x . (Not e that χ is w ell deﬁne d thanks to the c hoice of ε .) Let ξ t b e a sample path of the Bro wnian motio n. W e deﬁne inductiv ely the followin g Mark o v stopping times ( R n ) n ≥ 1 and ( S n ) n ≥ 0 as follows . Set S 0 = 0 if ξ 0 / ∈ X , and S 0 = min { t ≥ 0 , ξ t / ∈ V ξ 0 } . Then, for n ≥ 1, let  R n = min { t ≥ S n − 1 , ξ t ∈ F } S n = min { t ≥ R n , ξ t / ∈ V X n } with X n = χ ( ξ R n ). Let us also deﬁne recursiv ely for k ≥ 0 on Ξ × [0 , 1 ] N ,    N 0 ( ξ , α ) = 0 N k ( ξ , α ) = min { n > N k − 1 ( ξ , α ) , α n < κ n ( ξ ) } where κ n ( ξ ) = 1 C dε V X n dε V ξ R n ( ξ S n ) , and, for z ∈ F , ε V z denotes the distribution of ξ S 1 for sample paths ξ t starting at z . W e also set T k = S N k . F or y in M , we let e P y denote the pro duct measure of P y × λ N , where λ is the Leb esgue measure o n [0 , 1]. W e then deﬁne on X , the la w µ y ( x ) = e P y [ X N 1 = x ] . The f ollo wing pro p erties ar e kno wn to hold [40, 27, 3, 32]. Theorem 6.1. L et us deﬁne µ ( γ ) = µ w ( γ ( w )) , and Z k ( w ) = X N k with Z 0 ( w ) = w . (i) The r andom se quenc e ( Z n ( w )) is the r andom walk ge n er ate d by µ : for any x 1 = γ 1 ( w ) , . . . , x n = γ n ( w ) ∈ X , e P w ( Z 1 = x 1 , . . . , Z n = x n ) = µ ( γ 1 ) µ ( γ − 1 1 γ 2 ) . . . µ ( γ − 1 n − 1 γ n ) . (ii) The me asur e µ is symmetric with ful l supp ort but has a ﬁnite ﬁrst mom ent with r esp e ct to d . (iii) The Gr e en function G µ of the r ando m walk is pr op ortional to the Gr e en function G M of M . (iv) T her e exists a p ositive c onstant T such that the fol lowing limit ex i s ts a l m ost sur ely and in L 1 : lim S N k k = T . (v) Almost sur ely and in L 1 , lim d ( ξ k T , Z k ( w )) k = 0 . (vi) T he harmonic me asur es for the Br ownian motion and the r ando m walk c oincide. W e ar e able to prov e the following: 38 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU Theorem 6.2. Under the notation and assumptions fr om ab ove, let d G denote the Gr e en metric asso ciate d w ith µ . I f N is c omp act, then d G ∈ D (Γ) and dim ν = h M εℓ M wher e h M and ℓ M denote the entr opy and the drift of the B r ownian motion r esp e ctively. Proo f. The acron yms (ED) and (QR) below refer to Prop osition 3.5. Since M has pinc hed negativ e curv a ture, it follows that G M ( x, y ) . e − cd ( x,y ) holds for some constan t c > 0, see [2, (2.4) p. 434]. By part (iii) of Theorem 6.1, G µ and G M are prop ortiona l. Therefore G µ also satisﬁes G µ ( x, y ) . e − cd ( x,y ) and (ED) is prov ed. F urthermore, A. Ancona’s Theorem 3.7 also holds for the Br ownian motio n, see [1], showing that (QR) holds as well. Both these prop erties imply that ( X , d G ) ∈ D (Γ) by Prop osition 3.5. The identit y h µ = h M · T w as prov ed by V. Kaimanov ic h [2 5, 27]. F urthermore, f r om Theorem 6 .1 (4), it follows that almost surely , ℓ µ = lim d ( w , Z k ( w )) k = lim d ( w , ξ k T ) k = ℓ M · T . Th us, Corollar y 1.4 implies that dim ν = h M εℓ M . The computat io n o f the drift can also b e found in [3 2 ]. 6.2. E xp onen tial momen t for the discretised motion. In [1], A. Ancona w rote in a remark that the r a ndom walk deﬁned ab o v e has a ﬁnite exp onential momen t when N is compact. Since this fact is crucial to us, w e pro vide here a detailed pro of. This will enable us to apply Theorem 1.5 and conclude the pro of of Theorem 1.9. Theorem 6.3. If N is c omp act, then the r andom walk ( Z n ) deﬁne d in The or em 6.1 has a ﬁnite exp onential moment. The pro of requires in termediate estimates on the Brownian motion. The main step is an estimate on the p o sition of ξ S 1 : Prop osition 6.4. Ther e ar e p ositive c onstants C 1 and c 1 such that, for any r ≥ 1 , sup y ∈ M P y [ d ( ξ 0 , ξ S 1 ) ≥ r ] ≤ C 1 e − c 1 r . Prop osition 6.4 follow s from the follo wing lemma. Lemma 6.5. We write ξ ∗ t = sup 0 ≤ s ≤ t d ( ξ 0 , ξ s ) . Ther e ar e c onstants m > 0 , c 2 > 0 and C 2 > 0 such that sup y ∈ M P y [ ξ ∗ t ≥ mt ] ≤ C 2 e − c 2 t . Proo f. W e ﬁrst pro ve that all t he expo nen tial momen t s of ξ ∗ 1 are ﬁnite. Our pro of r elies on the f o llo wing upp er Gaussian estimate v a lid as so on as the curv ature is b ounded (see e.g. [4 5, § 6] for a proo f ): for any y ∈ M and an y t ≥ 2, P y [ ξ ∗ 1 ≥ t ] ≤ exp  − ct 2  , for some constan t c t ha t do es no t depend on y nor on t . HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 39 Hence, if λ > 0 then E y  e λξ ∗ 1  = 1 + Z u> 0 e u P y [ ξ ∗ 1 ≥ ( u/λ )] d u ≤ 1 + Z u> 0 e u − c u 2 λ 2 du < ∞ . Let y ∈ M and m > 0. It follo ws fro m the ex p onen tial Tc heb yc hev inequality that P y [ ξ ∗ t ≥ mt ] ≤ e − λmt E y  e λξ ∗ t  . W e remark that, for n ≥ 1 and t ∈ ( n − 1 , n ], ξ ∗ t ≤ X 0 ≤ k < n sup k ≤ s ≤ k +1 d ( ξ k , ξ s ) . It follows from t he Marko v prop erty that, f o r all y ∈ M , E y  e λξ ∗ t  ≤  sup z ∈ M E z  e λξ ∗ 1   n . Therefore P y [ ξ ∗ t ≥ mt ] . e − λmt  sup z ∈ M E z  e λξ ∗ 1   t . So, if m is c hosen large enough, w e will ﬁnd c 2 > 0 so that P y [ ξ ∗ t ≥ mt ] . e − c 2 t . Proo f of P roposition 6.4. The compactness of N easily implies the follo wing upper b ound on the ﬁrst hitting time S 1 using the orthog onal decomposition of L 2 ( N ) (se e [44, (5.2)]): there are p ositive constan ts C 3 and c 3 suc h that, for an y y ∈ M , (21) P y [ S 1 ≥ k ] ≤ C 3 e − c 3 k Let us consider κ > 0 that will b e ﬁxed later. P y [ d ( y , ξ S 1 ) ≥ r ] ≤ P y [ d ( y , ξ S 1 ) ≥ r ; S 1 ≤ κ ] + P y [ d ( y , ξ S 1 ) ≥ r ; S 1 ≥ κ ] . F rom (2 1), it follo ws that P y [ d ( y , ξ S 1 ) ≥ r ] . P y [ ξ ∗ κ ≥ r ] + e − c 3 κ . Cho osing κ = r /m , Lemma 6.5 implies that P y [ d ( y , ξ S 1 ) ≥ r ] . e − c 2 m r + e − c 3 m r , and the prop o sition fo llo ws. Proo f of Theorem 6.3. Let r ≥ 1 and k ≥ 1, and λ > 0 that will b e ﬁxed later. The exp o nen tial Tc heb yc hev inequalit y yields P y [ d ( ξ 0 , ξ S k ) ≥ r ] ≤ e − λr E y  e λd ( ξ 0 ,ξ S k )  . But d ( ξ 0 , ξ S k ) ≤ X 0 ≤ j 0 e u P y [ d ( ξ 0 , ξ S 1 ) ≥ ( u/λ )] du ≤ 1 + C 1 Z u> 0 e u e − c 1 u/λ du . W e choose λ < c 1 ; there exists a positive constan t C 4 suc h that sup z ∈ M E z  e λd ( ξ 0 ,ξ S 1 )  ≤ 1 + C 4 λ 1 − ( λ/c 1 ) . Plugging this last inequality in (22) yields (23) P y [ d ( ξ 0 , ξ S k ) ≥ r ] . e − λr + k c 4 λ for some constan t c 4 > 0. W e not e that, for an y x ∈ X , an y z ∈ F x and u ∈ ∂ V x , dε V x dε V z ( u ) ≥ (1 /C ) where C is the Harnac k constan t. Observ e that this estimate is uniform with respect to u ∈ ∂ V x and z ∈ F x . Therefore, e P y [ T 1 ≥ S k | ξ ] = e P y  ∩ k − 1 n =1 { κ n ( ξ ) < α n }| ξ  ≤ e P y  ∩ k − 1 n =1 { (1 /C 2 ) < α n }| ξ  = e P y  ∩ k − 1 n =1 { (1 /C 2 ) < α n }  = k − 1 Y n =1 e P y [(1 /C 2 ) < α n ] . (1 − (1 / C 2 )) k . (24) In (24 ), w e used the nota tion e P y [ . | ξ ] to denote the conditio nal probabilit y giv en the Bro w- nian pa th ξ . Not e that S k , b eing a f unction of ξ , do es not depend on the sequence α . W e used this fact for the second equalit y a b ov e; see also [32] for a diﬀeren t argumen t leading to the same conclusion. HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 41 F rom (2 3) and (24), it then follows that e P y [ d ( y , ξ T 1 ) ≥ r ] = X k ≥ 1 e P y [ d ( y , ξ S k ) ≥ r ; S k = T 1 ] = X k ≥ 1 e E y [ e P y [ S k = T 1 | ξ ] ; d ( y , ξ S k ) ≥ r ] . X k ≥ 1 (1 − (1 /C 2 )) k e P y [ d ( y , ξ S k ) ≥ r ] . e − λr X k ≥ 1 (1 − (1 /C 2 )) k e λc 4 k . Th us, there is some λ 0 > 0 so that if w e choose λ ∈ (0 , λ 0 ] then this last serie s is con v ergen t and w e ﬁnd e P y [ d ( y , ξ T 1 ) ≥ r ] . e − λr . Consequen tly , noting that d ( Z 1 ( w ) , ξ T 1 ) ≤ ε and c ho o sing λ = λ 0 , E  e ( λ 0 / 2) d ( y,Z 1 ( w ))  . 1 + Z u> 0 e u e P [ d ( y , ξ T 1 ) > 2 u/λ 0 ] du . 1 + Z u> 0 e − u du < ∞ . 6.3. E xamples. Let us ﬁx n ≥ 2 and conside r the hy p erb olic space H n of constan t sectional curv ature − 1. The explicit form of the Green function on this space sho ws easily that, giv en w , x, y , z ∈ H n whic h are at distance c > 0 apart from one another, one has (25) Θ( x, y ) & min { Θ ( x, z ) , Θ( z , y ) } where Θ is Na ¨ ım’s k ernel, and the implicit constan t dep ends only on c . Let N b e a ﬁnite v olume hyperb olic manifold with dec k transformat io n g roup Γ acting on H n . The estimate (25) sho ws that the Green metric d G on Γ asso ciated with the discretised Brownian motion on H n is hyperb olic. Moreo v er, the estimate (ED) holds as well, so that the G reen metric d G is quasi-isome tric to the restriction of the hyperb olic metric to the or bit Γ( o ) of a base point o ∈ H n . Since N has ﬁnite volume , the limit set of Γ is the whole sphere at inﬁnit y , a nd it coincides with the visual bo undary of ( Γ , d G ). Therefore, Theorem 1.7 implies tha t the Martin b oundary coincides with ∂ H n , homeomorphic to S n − 1 . W e omit the details. W e apply this cons truction in t w o special cases . If w e consider for N a punctured 2- torus with a complete h yperb olic metric of ﬁnite v olume (as in [3]), w e obtain an example of a random w alk on the free group for whic h the Green metric is h yperb olic but its b o undary S 1 do es not coincide with the b oundary of the group (whic h is a Can tor se t). T herefore, d G do es not b elong to the quasi-isometry class of the free group. If w e consider no w for N a complete h yp erb olic 3-manifold of ﬁnite volume with a rank 2 cusp, then its fundamental group is not h yp erb olic since it con tains a subgroup isomorphic to Z 2 , but the Green metric is h yp erb olic no netheless. 42 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU Appendix A. Quasiruled hyp e rbolic sp aces F or geo desic spaces, hy p erb olicity admits many c haracterisations based on geo desic tria ngles (cf. Prop. 2.21 from [22 ]). Most of them still hold when the space X is j ust a length space (see eg. [47]) . F or instance, a geo desic h yperb o lic space satisﬁes R ips condition, namely , a constan t δ exists suc h that any edge of a geo desic triangle is at distance a t most δ from the t w o other edges. It is kno wn that if X and Y are t w o quasi-isometric geo desic spaces , then X is h yperb o lic if and only if Y is (Theorem 5.12 in [22]). This statemen t is known to b e false in general if w e do not assume b o th spaces to b e geodesic (Example 5.12 f rom [22], and Prop osition A.11 b elo w). Since quasi-isometries do not preserv e small-scales of metric spaces , in particular geo desics, it is therefore imp ortant to ﬁnd o t her coa r se c ha r acterisations of hy p erb olicity . Suc h a c har- acterisation is the purp ose of this app endix. W e prop o se a setting whic h enables us to go through the whole t heory of quasiconformal measures as if the underlying space w as g eo desic. Deﬁnition. A quasige o desic curve (r esp. r ay, se gment) is the image of R (res p. R + , a compact in terv al of R ) b y a quas i-isometric em b edding. A space is said to b e quasige o desic if there are constan ts λ , c suc h that an y pair of p oin ts can b e connected b y a ( λ, c )- quasigeo desic. The image of a geo desic space by a quasi-isometry is thus quasigeo desic. But as it w as men tioned earlier, h yperb olicit y need not be preserv ed. Deﬁnition. A τ -quasiruler is a quasigeo desic g : R → X (resp. quasisegmen t g : I → X , quasira y g : R + → X ) suc h that, for an y s < t < u , ( g ( s ) | g ( u )) g ( t ) ≤ τ . Let X b e a metric space. Let λ ≥ 1 and τ , c > 0 b e constan ts. A quasiruling structur e G is a set of τ -quasiruled ( λ , c ) -quasigeo desics such any pair of points of X can b e joined b y an elemen t of G . A metric space will b e quasirule d if constan ts ( λ, c, τ ) exist so that the space is ( λ, c )- quasigeo desic and if ev ery ( λ, c )-quasigeo desic is a τ -quasiruler i.e., the set of quasigeodesics de- ﬁnes a quasirulin g structure. The data of a quasiruled space ar e th us the constan ts ( λ, c ) for the quasigeo desics and the constan t τ giv en b y the quasiruler prop erty of the ( λ, c )- quasigeo desics. A quasi-isometric em b edding f : X → Y b et w een a geo desic metric space X into a metric space Y is τ -ruling if the image of any geo desic segmen t is a τ -quasiruler. Then the ima g es of geo desics of X deﬁne a quasiruling structure G of Y . In this situation, we will sa y that G is induced b y X . Theorem A.1. L et X b e a ge o desic hyp erb olic me tric sp ac e, a n d ϕ : X → Y a quasi-isom etry, wher e Y is a metric sp ac e. The f o l lowing statements ar e e quivalent: (i) Y is hyp erb olic; (ii) Y is quasirule d; (iii) ϕ is ruling. Mor e o v er if Y is a hyp erb olic quasirule d sp ac e, then Y is isometric to a quasic onv e x s ubset of a ge o desic hyp erb olic metric sp ac e Z . F urthermor e, if Γ acts ge ometric al ly on Y , then Γ is a quasic o nvex gr oup acting on Z . HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 43 Theorem 1 .10 is a consequence from Theorem A.1. W e refer to [22] for any undeﬁned not io n used in the seq uel. A.1. Straigh tening of conﬁgurations. Let I = [ a, b ] ⊂ R b e a closed connected subset. W e assume throughout this section that constan ts ( λ, c, τ ) are ﬁxed. Lemma A.2. L et g : I → X b e a quasiruler. Ther e is a (1 , c 1 ) -quasi-isometry f : g ( I ) → [0 , | g ( b ) − g ( a ) | ] , for some c 1 which dep end s only on the data ( λ , c and τ ). Proo f. F or an y x ∈ g ( I ), let f ( x ) = min {| x − g ( a ) | , | g ( b ) − g ( a ) | } . Th us (26) || x − g ( a ) | − f ( x ) | ≤ 2 τ . Let x, y ∈ g ( I ) with x = g ( s ) and y = g ( t ), and let us assume that s < t . • W e apply ( 2 6) rep eatedly . On the one hand, | f ( x ) − f ( y ) | ≤ || x − g ( a ) | − | y − g ( a ) || + 4 τ ≤ | x − y | + 4 τ . On t he other hand, since s < t , it follow s tha t | x − g ( a ) | + | x − y | ≤ | y − g ( a ) | + 2 τ so that | f ( x ) − f ( y ) | ≥ | x − y | − 8 τ . Hence f is a ( 1 , 8 τ )-quasi-isometric em b edding. Note that the constan ts ab ov e are not sharp (a case b y case treatmen t would divide most of them by 2). • If | a − b | ≤ 2, then | f ( g ( a )) − f ( g ( b )) | = | g ( a ) − g ( b ) | ≤ 2 λ + c and f is cobo unded. Otherwise, | a − b | > 2. Let s j = a + j for j ∈ N ∩ [0 , | b − a | ]. It follo ws that | f ( g ( s j )) − f ( g ( s j +1 )) | ≤ λ | s j − s j +1 | + c + 4 τ ≤ λ + c + 4 τ . The set { f ( g ( s j )) } j is a c hain in [0 , | g ( b ) − g ( a ) | ] whic h joins 0 to | f ( g ( a )) − f ( g ( b )) | = | g ( b ) − g ( a ) | ; since t w o consecutiv e p oints of { f ( g ( s j )) } j are at most λ + c + 4 τ apart, it follows that its ( λ + c + 4 τ )-neigh b orho o d co vers [0 , | g ( a ) − g ( b ) | ], hence f is a quasi- isometry . Remark. If f a denotes the map as abov e and f b : g ( I ) → [0 , | g ( b ) − g ( a ) | ] the map suc h that f b ( g ( b )) = 0 , then | f a ( x ) + f b ( x ) − | g ( a ) − g ( b ) || ≤ 2 τ holds. Deﬁnition. Giv en three p oin ts { x, y , z } , there is a trip o d T and an isometric embedding f : { x, y , z } → T suc h that the images are the endpoints of T . W e let ¯ c denote the cen ter of T . A quasitriangle ∆ is giv en b y three p oin ts x, y , z together with three quasirulers j o ining them. W e will denote the edges b y [ x, y ], [ x, z ] and [ y , z ]. Suc h a quasitriangle is δ - t hin if an y segmen t is in the δ -neigh b orho o d o f the tw o others. Lemma A .3. L et ∆ b e a δ -thin quasitriangle with vertic es { x, y , z } . Ther e is a (1 , c 2 ) -quasi- isometry f ∆ : ∆ → T , wher e T is the trip o d asso ciate d with { x, y , z } and c 2 dep ends only on the data ( δ , λ , c , τ ). 44 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU Proo f. Let us deﬁne f ∆ using Lemma A.2 on eac h edge. This map is clearly cob ounded. Let u, v ∈ ∆. Since ∆ is thin, one ma y ﬁnd tw o p oin ts u ′ , v ′ ∈ ∆ on the same edge suc h that | u − u ′ | ≤ δ and | v − v ′ | ≤ δ , so that || u − v | − | u ′ − v ′ || ≤ 2 δ . If u and u ′ b elong to the same edge, then | f ∆ ( u ) − f ∆ ( u ′ ) | ≤ | u − u ′ | + c 1 ≤ δ + c 1 . Otherwise, let x b e the common vertex of the edges con taining u and u ′ , then it follows from (26) tha t | f x ( u ) − f x ( u ′ ) | ≤ | u − u ′ | + 4 τ ≤ δ + 4 τ and similarly for v and v ′ . Th us | f ∆ ( u ) − f ∆ ( u ′ ) | , | f ∆ ( v ) − f ∆ ( v ′ ) | ≤ c ′ , where c ′ dep ends only on the data. It f o llo ws that || f ∆ ( u ) − f ∆ ( v ) | − | f ∆ ( u ′ ) − f ∆ ( v ′ ) || ≤ 2 c ′ . But since u ′ and v ′ b elong to the same edge, Lemma A.2 implies that || f ∆ ( u ′ ) − f ∆ ( v ′ ) | − | u ′ − v ′ || ≤ c 1 , so || f ∆ ( u ) − f ∆ ( v ) | − | u ′ − v ′ || ≤ 2 c ′ + c 1 and ﬁnally || f ∆ ( u ) − f ∆ ( v ) | − | u − v || ≤ (2 c ′ + c 1 + 2 δ ) . In the situation of Lemma A.3 w e ha v e | ( f ∆ ( x ) | f ∆ ( y )) f ∆ ( z ) − ( x | y ) z | ≤ C , for some unive rsal constant C > 0; th us, w e ma y ﬁnd po in ts c x ∈ [ y , z ], c y ∈ [ x, z ] and c z ∈ [ y , x ] suc h tha t | f ∆ ( c x ) − ¯ c | , | f ∆ ( c y ) − ¯ c | , | f ∆ ( c z ) − ¯ c | ≤ c 3 , and diam { c x , c y , c z } ≤ c 3 , where c 3 dep ends only on the data. Prop osition A.4. L et X b e a metric sp ac e endowe d with a quasiruling structu r e G such that al l quasitriangles ar e δ -thin. Then X is hyp erb olic quantitatively: the c onstant of hyp e rb olicity only dep ends on ( δ, λ, c, τ ) . Proo f. Let us ﬁx w , x, y , z ∈ X . Let us consider the followin g triangles : A = { w , x, z } and B = { w , x, y } . Let us denote by T A , T B and ¯ c A , ¯ c B the a sso ciated trip o d a nd center resp ectiv ely , and let us deﬁne Q = T A ∪ T B where b oth copies f A ([ w , x ]) and f B ([ w , x ]) of [ w , x ] hav e b een iden tiﬁed. This metric space Q is to p ologically an “ × ”, and so is of course 0-h yp erb olic. Let us deﬁne f : A ∪ B → Q b y sending A under f A and B under f B . The r estriction of f to A and to B is a (1 , c 2 )-quasi-isometry b y Lemma A.3 . HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 45 It f o llo ws that | f ( y ) − f ( z ) | = | f ( y ) − ¯ c B | + | ¯ c B − ¯ c A | + | ¯ c A − f ( z ) | . One ma y ﬁnd c A , c B ∈ [ w , x ] suc h that | f ( c A ) − ¯ c A | ≤ c 3 and | f ( c B ) − ¯ c B | ≤ c 3 . Lemma A.3 implies that | f ( y ) − f ( c B ) | = | y − c B | and | f ( c A ) − f ( z ) | = | c A − z | up to an additiv e constant. Therefore, | f ( y ) − ¯ c B | = | y − c B | and | ¯ c A − f ( z ) | = | c A − z | up to an additiv e constant to o . By Lemma A.2, | ¯ c B − ¯ c A | = | c B − c A | up to an additiv e constant, whence the existence of some constan t c 4 > 0 suc h that | f ( y ) − f ( z ) | ≥ | y − c B | + | c B − c A | + | c A − z | − c 4 ≥ | y − z | − c 4 . Hence ( f ( y ) | f ( z )) f ( w ) ≤ ( y | z ) w + c 4 . It follo ws from the h yp erb olicit y of Q t ha t ( y | z ) w ≥ min { ( f ( x ) | f ( z ) ) f ( w ) , ( f ( y ) | f ( x )) f ( w ) } − c 4 and since the restrictions of f to A and B are (1 , c 2 )-quasi-isometries, min { ( f ( x ) | f ( z )) f ( w ) , ( f ( y ) | f ( x )) f ( w ) } − c 4 ≥ min { ( x | z ) w , ( y | x ) w } − c 5 for some constan t c 5 . W e ha ve just establis hed that for an y w , x, y , z , ( y | z ) w ≥ min { ( x | z ) w , ( y | x ) w } − c 5 . A.2. Em b eddings of hy p erb olic spaces. W e recall a theorem of M. Bonk and O. Sc hramm (Theorem 4 .1 in [10]) : Theorem A.5. Any δ -hyp erb olic sp ac e X c an b e isometric al ly emb e dde d in to a c omplete ge o desic δ -hyp erb olic sp ac e Y . W e will show that if Γ acts isometrically on X , then so is the case on Y . T o prov e this w e need to review the construction of the set Y . The ﬁr st lemma, whic h w e recall, is the basic step in the construction. Lemma A.6. L et X b e δ -hyp e rb olic metric sp ac e, and let a 6 = b b e in X . I f, f o r every x , ( | a − b | / 2 , | a − b | / 2) 6 = ( | a − x | , | b − x | ) , then ther e is a δ -hyp erb olic sp ac e X [ a, b ] = X ∪ { m } such that ( | a − b | / 2 , | a − b | / 2) = ( | a − m | , | b − m | ) . F urthermor e, for any x ∈ X , | x − m | = | a − b | 2 + sup w ∈ X ( | x − w | − max {| a − w | , | b − w |} ) . W e call m the middle p oin t of { a, b } . Lemma A.7. A δ -hyp e rb olic metric sp ac e X em b e ds isome tric al ly into a δ -h yp erb olic sp ac e X ∗ such that, for any ( a, b ) ∈ X , ther e exists a midd le p oint m = m ( a, b ) ∈ X ∗ . Proo f. The y apply a transﬁnite induction : let φ : ω → X × X b e an ordinal o f X × X . Deﬁne indu ctiv ely X ( α ) as follow s. Set X (0) = X . If α = β + 1 ≤ ω + 1, then deﬁne X ( α ) = X ( β )[ φ ( α )]. Clearly , X ( α ) is δ - h yp erb olic. If α is a limit ordinal, set X ( α ) = ( ∪ β <α X ( β )) [ φ ( α )] . Here to o , X ( α ) is δ -hyperb o lic since δ -h yp erb olicity is preserv ed under increasing unions. The space X ∗ = X ( ω + 1) fulﬁlls the req uiremen ts. F or α ≤ ω + 1, let us deﬁne m ( α ) = m ( φ ( α )) the middle of φ ( α ) = ( a ( α ) , b ( α )), and let D ( α ) = | a − b | . If x ∗ ∈ X , set α ( x ∗ ) = 0 ; otherwise, let P ( x ∗ ) b e the set of ordinals α suc h that x ∗ ∈ X ( α ). Let us deﬁne α ( x ∗ ) as the minim um of P ( x ∗ ); it follo ws tha t x ∗ = m ( α ). W e let D ( x ∗ ) = D ( α ). W e also write φ ( α ) = ( a ( x ∗ ) , b ( x ∗ )). 46 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU Lemma A.8. L et α < β , then | m ( α ) − m ( β ) | = D ( β ) 2 + sup w ∈ X ( α ) {| w − m ( α ) | − max {| w − a ( β ) | , | w − b ( β ) |}} . Proo f. Let Z = ( γ ∈ ω , | m ( α ) − m ( γ ) | = D ( γ ) 2 + sup w ∈ X ( α ) {| w − m ( α ) | − max {| w − a ( γ ) | , | w − b ( γ ) |}} ) . The set Z con tains { γ ≤ α + 1 } b y deﬁnition. Let us assume that { γ < β } ⊂ Z for some β > α . Pic k γ ∈ Z , so that α < γ < β . Giv en ε > 0, there is some w ∈ X ( α ) so that | m ( α ) − m ( γ ) | ≤ D ( γ ) 2 + | w − m ( α ) | − ma x {| w − a ( γ ) | , | w − b ( γ ) |} + ε . Since w ∈ X ( α ) is ﬁxed, | m ( γ ) − a ( β ) | ≥ D ( γ ) 2 + | w − a ( β ) | − max {| w − a ( γ ) | , | w − b ( γ ) |} . A similar statemen t holds for b ( β ) instead of a ( β ). Therefore max {| m ( γ ) − a ( β ) | , | m ( γ ) − b ( β ) |} ≥ D ( γ ) 2 + max {| w − a ( β ) | , | w − b ( β ) |} − max {| w − a ( γ ) | , | w − b ( γ ) |} , and | m ( α ) − m ( γ ) | − max {| m ( γ ) − a ( β ) | , | m ( γ ) − b ( β ) |} ≤ | m ( α ) − w | − max {| w − a ( β ) | , | w − b ( β ) |} + ε . It fo llows that, fo r each α < γ < β and ε > 0 , there is some w ∈ X ( α ) suc h that the suprem um in the deﬁnition of | m ( α ) − m ( β ) | is attained within X ( α ) up to ε . He nce β ∈ Z , so Z = X ∗ b y induction. Lemma A.9. L et 0 < α < β . Then | m ( α ) − m ( β ) | c an b e c ompute d as D ( α ) 2 + D ( β ) 2 + sup w ,w ′ ∈ X {| w − w ′ | − (max {| w − a ( α ) | , | w − b ( α ) | } + max {| w ′ − a ( β ) | , | w ′ − b ( β ) |} ) } . Proo f. W e endow ω × ω with the lexicographical order, and w e consider ω ′ = { ( α , β ) , α < β } . W e assume by transﬁnite induction that the lemma is true for any ( α , β ) < ( b α, b β ). By Lemma A.8, give n ε > 0, there is some b w ∈ X ( b α ) suc h that | m ( b α ) − m ( b β ) | ≤ D ( b β ) 2 + | b w − m ( b α ) | − max {| b w − a ( b β ) | , | b w − b ( b β ) |} + ε . It follows from t he induction assumption that there are p oin ts w ′ , w ∈ X suc h that | m ( b α ) − b w | − ε ≤ D ( b α ) 2 + D ( b w ) 2 + | w − w ′ | − (max {| w − a ( b α ) | , | w − b ( b α ) | } + max {| w ′ − a ( b w ) | , | w ′ − b ( b w ) | } ) . But max {| b w − a ( b β ) | , | b w − b ( b β ) |} ≥ D ( b w ) 2 + max {| w ′ − a ( b β ) | , | w ′ − b ( b β ) |} − max {| w ′ − a ( b w ) | , | w ′ − b ( b w ) |} , HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 47 so | m ( b α ) − m ( b β ) | − 2 ε ≤ D ( b α ) 2 + D ( b β ) 2 + | w − w ′ | − (max {| w − a ( b α ) | , | w − b ( b α ) | } + max {| w ′ − a ( b β ) | , | w ′ − b ( b β ) |} ) . This establishes the lemma. Corollary A .10. If Γ acts on X by isometry, then it acts also on X ∗ by isometry. Proo f. If x ∗ ∈ X ∗ \ X and g ∈ Γ, w e let g ( x ∗ ) = m ( g ( a ( x ∗ )) , g ( b ( x ∗ )))). The fact tha t g : X ∗ → X ∗ acts b y isometry follows from Lemma A.9 since the distance b et w een t w o p oin ts relies o nly on points inside X . The construction now go es a s fo llows. D eﬁne X 0 = X , a nd X n +1 = X ∗ n , for n ≥ 0. The space X ′ = ∪ n ∈ N X n is a metric δ -hyperb olic space such that any pair o f p oints admits a midp oin t in X ′ . Note that if Γ acts on X b y isometry , then it also acts b y isometry on X ′ . T o obta in a complete geo desic space, M. Bo nk a nd O. Sc hramm use again a transﬁnite induction. Let ω 0 b e the ﬁrst uncoun table or dina l. They deﬁ ne a metric space Z ( α ) for eac h ordinal α < ω 0 suc h that Z ( α ) ⊃ Z ( β ) if α > β . W e se t Z (0) as the completion of X ′ . More generally , if α = β + 1, deﬁne Z ( α ) as the completion of Z ( β ) ′ . F or limit ordinals α , w e deﬁne Z ( α ) as the completion of ∪ β <α Z ( β ) ′ . It follo ws that for eac h α < ω 0 , the metric sp ace Z ( α ) is complete, δ -hyperb o lic, and a dmits an isometric action of Γ if X did. The construction is completed b y letting Y = ∪ α<ω 0 Z ( α ). As ab ov e, an action of a group Γ by isometry on X extends canonically as a n a ctio n b y isometry on Y . A.3. Quasiruled spaces and h yp erb olicity. W e pro ve Theorem A.1 in four steps. A.3.1. Let us assume that Y is a quasigeo desic δ -h yp erb olic space. It fo llows from Theorem A.5 that there are a δ -hy p erb olic geo desic metric space b Y and an isometric em b edding ι : Y → b Y . Thu s, for an y quas igeo desic segmen t g : [ a, b ] → Y , ι ( g ) shado ws a gen uine geo desic b g = [ ι ( g ( a )) , ι ( g ( b ))] from b Y at distance H = H ( λ, c, δ ). In other w ords, for an y t ∈ [ a, b ], there is a p oint b y t ∈ b g suc h that | ι ( g ( t )) − b y t | ≤ H . It f o llo ws that ( g ( a ) | g ( b )) g ( t ) ≤ ( ι ( g ( a )) | ι ( g ( b )) b y t + H = H since ι ( g ( a )), ι ( g ( b )) and b y t b elong to a geo desic segmen t. Therefore, Y is quasiruled. A.3.2. If Y is quasiruled, then ϕ is ruling sinc e the image under ϕ is a quasigeo desic, hence a quasiruler b y deﬁnition. A.3.3. Let us no w assume that X is a geo desic h yp erb olic space and ϕ : X → Y is a quasi- isometry in to a metric space Y . It follows that Y is quasigeo desic and that the edge of the image of any geo desic triangle is at a bo unded distance from the t w o other edges i.e., triangles are δ -thin. If ϕ is ruling, then Propo sition A.4 applies, and pro v es that Y is hyperb olic. A.3.4. The statemen t concerning group actions follows from ab ov e and the previous section. 48 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU A.4. Non-h yp erb olic in v arian t metric on a hyperb olic group. In [22], the a uthors pro vide an example of a non- h yp erb olic metric space quasi-isometric to R . One could w onder if, in the case of gro ups, the in v ariance of hyperb o licit y holds f or quasi-isometric and inv ariant metrics. In this section, w e dispro v e this statemen t. Prop osition A.11. F or any hyp erb olic gr oup, a left-invariant metric quasi- i s ometric to a wor d metric exists which is not hyp erb olic. W e are grateful to C. Pittet and I. Miney ev fo r ha ving p oin ted out to us the metric d in the follo wing pr o of as a p ossible candidate. Proo f. Let Γ b e a h yp erb olic group and let | . | denote a word metric. W e deﬁne the metric d ( x, y ) = | x − y | + log(1 + | x − y | ) . Clearly , | x − y | ≤ d ( x, y ) ≤ 2 | x − y | holds and d is left-in v arian t b y Γ. Let us pro v e that (Γ , d ) is not quasiruled, hence not h yp erb olic b y Theorem A.1. Let g b e a geodesic for | . | whic h w e iden tify with Z . Since (Γ , d ) is bi-Lipsc hitz to (Γ , | · | ), it is a (2 , 0)- quasigeo desic for d . But d (0 , n ) + d ( n, 2 n ) − d (0 , 2 n ) = log (1 + n ) 2 / (1 + 2 n ) asymptotically behav es as log n . Therefore g is not quasiruled. Appendix B. App ro xima te trees and shadows Appro ximate trees is an imp ortant to ol to understand h yp erb olicit y in geo desic spaces. Here, w e a dapt their existe nce to the setting of hy p erb olic quasiruled metric spaces following E. Ghy s and P . de la Harp e (Theorem 2.12 in [22]). Theorem B.1. L et ( X , w ) b e a δ -hyp erb olic metric sp ac e and let k ≥ 0 . (i) If | X | ≤ 2 k + 2 , then ther e is a ﬁnite m e tric p oi nte d tr e e T and a map φ : X → T such that : → ∀ x ∈ X , | φ ( x ) − φ ( w ) | = | x − w | , → ∀ x, y ∈ X , | x − y | − 2 k δ ≤ | φ ( x ) − φ ( y ) | ≤ | x − y | . (ii) If ther e ar e τ -quasirule d r ays ( X i , w i ) 1 ≤ i ≤ n with n ≤ 2 k such that X = ∪ X i , then ther e is a p o i n te d R -tr e e T and a map φ : X → T such that → ∀ x ∈ X , | φ ( x ) − φ ( w ) | = | x − w | , → ∀ x, y ∈ X , | x − y | − 2( k + 1) δ − 4 c − 2 τ ≤ | φ ( x ) − φ ( y ) | ≤ | x − y | , wher e c = max {| w − w i |} . W e rep eat the argumen ts in [22]. T he pro o fs of the ﬁrst t wo lemmata can b e found in [22], and the last one is the quasiruled v ersion of [22, Lem. 2.1 4 ]. In the three lemmata, X is assumed to b e δ -h yp erb olic. F urthermore, w e will omit the subscript w for the inner pro duct and write ( ·|· ) = ( · |· ) w . Lemma B.2. We d eﬁne → ( x | y ) ′ = sup min { ( x i − 1 | x i ) , 2 ≤ i ≤ L } , wher e the supr emum is taken over al l ﬁnite chains x 1 , . . . , x L with x 1 = x and x L = y , → | x − y | ′ = | x − w | + | y − w | − 2( x | y ) ′ , → x ∼ y if | x − y | ′ = 0 . HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 49 Then ∼ is an e quivalenc e r elation and | · | ′ is a distanc e on X/ ∼ w h ich ma k es it a 0 -hyp erb o l i c sp ac e. Mor e over, for an y x ∈ X , | x − w | ′ = | x − w | holds, and for any x, y ∈ X , | x − y | ′ ≤ | x − y | . Lemma B.3. I f | X | ≤ 2 k + 2 then for any chain x 1 , . . . , x L ∈ X , ( x 1 | x L ) ≥ min 2 ≤ j ≤ L { ( x j − 1 | x j ) } − k δ , holds. Lemma B.4. L et X = ∪ n i =1 X i wher e ( X i , w i ) ar e τ -q uasirule d r a ys. If n ≤ 2 k then, for an y chain x 1 , . . . , x L ∈ X , ( x 1 | x L ) ≥ min 2 ≤ j ≤ L { ( x j − 1 | x j ) } − ( k + 1) δ − 2 c − τ . Proo f. First, ( x | y ) w ≤ min {| x − w | , | y − w |} holds for any x, y ∈ X , and if x, y ∈ X i then | ( x | y ) w i − min {| x − w i | , | y − w i |}| ≤ τ , and | x − w i | ≥ | x − w | − | w − w i | ≥ | x − w | − c . Similarly , | y − w i | ≥ | y − w | − c . Th us, ( x | y ) w i ≥ min {| x − w | , | y − w |} − c − τ and ( x | y ) w ≥ ( x | y ) w i − c ≥ min {| x − w | , | y − w | } − 2 c − τ ≥ min { ( x | x ′ ) w , ( y | y ′ ) w } − 2 c − τ for all x ′ , y ′ ∈ X . Let x 1 , . . . , x L ∈ X b e a c hain. W e will write X ( x j ) to denote the quasiruled ray X i whic h con tains x j . Either, for all j ≥ 2, x j 6∈ X ( x 1 ), or there is a maximal index j > 1 suc h that x j ∈ X ( x 1 ). Hence, it follo ws from ab o v e that ( x 1 | x j ) ≥ min 2 ≤ i ≤ j { ( x i − 1 | x i ) } − 2 c − τ . In t his case, let us consider x 1 , x j , x j +1 , . . . , x L . W e inductiv ely extract a c hain ( x ′ i ) of length at most 2 n ≤ 2 k +1 whic h con tains x 1 and x L and suc h t hat at most tw o elemen ts b elong to a common X i , and in this case, they ha v e success iv e indices. It follo ws from Lemma B.3 and from ab o v e that ( x 1 | x L ) ≥ min { ( x ′ i − 1 | x ′ i ) } − ( k + 1) δ ≥ min { ( x i − 1 | x i ) } − ( k + 1) δ − 2 c − τ . Proo f of Theorem B.1. The t heorem follows as so on as w e ha v e found a quasi-isometric em bedding φ : X → T with T 0-h yperb olic. Lemma B.2 implies that X/ ∼ is 0-hyperb olic and that φ : X → X/ ∼ satisﬁes | φ ( x ) − φ ( w ) | ′ = | x − w | a nd | φ ( x ) − φ ( y ) | ′ ≤ | x − y | . F or cas e (i), Lemme B.3 sho ws that ( x | y ) ≥ ( x | y ) ′ − k δ i.e., | φ ( x ) − φ ( y ) | ′ ≥ | x − y | − 2 k δ. F or cas e (ii), Lemme B.4 sho ws that ( x | y ) ≥ ( x | y ) ′ − ( k + 1) δ − 2 c − τ i.e., | φ ( x ) − φ ( y ) | ′ ≥ | x − y | − 2( k + 1) δ − 4 c − 2 τ . Visual qua siruling structures. Let ( X, d, w ) b e a h yp erb olic space endow ed with a quasir- uling structure G . W e sa y that G is visual if any pair o f p oin ts in X ∪ ∂ X can b e joined b y a τ -quasiruled ( λ, c )- quasigeo desic. If X is a prop er space, then a ny quasiruling structure can b e completed into a visual quasiruling structure. Also, if Y is a h yp erb olic geo desic prop er metric space a nd ϕ : Y → X is ruling, then the induced quasiruling structure is also visual. This fact can in particular b e applied when Y is a locally ﬁnite Ca yley graph of a non-elemen tary h yperb olic group Γ, ( X, d ) ∈ D (Γ) a nd ϕ is the iden tit y map. Thus one endo ws ( X, d ) with a visual quasiruling structure. 50 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU Shado ws. Let ( X , d, w ) b e a h yperb olic quasiruled space endow ed with a visual quasiruling structure G . W e already deﬁned the shadow ℧ ( x, R ) in Section 2 as the set of points a ∈ ∂ X suc h that ( a | x ) w ≥ d ( w , x ) − R . An alternativ e deﬁnition is: let ℧ G ( x, R ) b e the set of p oints a ∈ ∂ X suc h that there is a quasiruler [ w , a ) ∈ G which in tersects B ( x, R ) = { y ∈ X : d ( x, y ) < R } . The following holds by applying Theorem B.1, since G is visual. Prop osition B.5. L et X b e a hyp erb olic sp ac e endowe d with a vi s ual quasiruling structur e G . Ther e exist p ositive c onstants C, R 0 such that for any R > R 0 , a ∈ ∂ X an d x ∈ [ w , a ) ∈ G , ℧ G ( x, R − C ) ⊂ ℧ ( x, R ) ⊂ ℧ G ( x, R + C ) . The whole theory of quasiconformal measures for h yp erb olic groups acting on geo desic spaces in [15] is based on the existence of appro ximate tr ees. Therefore, the same pro o f as in [15] leads to Theorem 2.3 and Lemma 2.4. Since quasiconformal measures are Ahlfors-regular, the lemma of the shado w also holds for shadows deﬁned b y visual quasiruling structures. Note that, in a hy p erb olic space endow ed with a visual quas iruling structure, Theorem B.1 implies that the deﬁnition of Busemann functions we gav e in Section 2 is equiv a len t to the classical o ne giv en b elow: Busemann functions. Let us a ssume that ( X , w ) is a po in ted h yp erb olic quasiruled space. Let a ∈ ∂ X , x, y ∈ X a nd h : R + → X a quasiruled ra y suc h that h (0 ) = y a nd lim ∞ h = a . W e deﬁne β a ( x, h ) def. = lim sup( | x − h ( t ) | − | y − h ( t ) | ) and β a ( x, y ) def. = sup { β a ( x, h ) , with h as ab ov e } . One can actually retriev e the metric from the Busemann functions as the next Lemma sho ws. Lemma B.6. L et ( X, w ) b e a p ointe d hyp erb o l i c quasirule d sp ac e w ith the fo l lowing quasi- starlike pr o p erty: ther e exists R 1 such that an y x ∈ X is at distanc e at most R 1 fr om a quasir ay [ w , a ) , a ∈ ∂ X . Then ther e exists a c onstant c 6 such that || x − y | − sup a ∈ ∂ X β a ( x, y ) | ≤ c 6 , for al l x, y ∈ X . The c o n stant c 6 dep ends only on the data ( δ , λ , c , τ ) . Observ e tha t the quasi-starlik e prop erty is satisﬁed as so on as there is a geometric group action on X . Proo f. F rom the tria ngle inequalit y , w e alwa ys ha v e β a ( x, y ) ≤ | x − y | . No w c ho ose x, y ∈ X and a ∈ ∂ X suc h that y is at distance at most R 1 from a quasira y [ w , a ). If these four points w ere really sitting on a tree, w e w ould ha v e β a ( x, y ) ≥ | x − y | − R 1 . Using a ppro ximate trees as in Theorem B.1, w e get the Lemma. Reference s [1] A. Ancona , Th´ eorie du p otentiel sur les g raphes et les v ari´ et ´ es, ´ Ec ole d’ ´ et´ e de Pr ob abilit´ es de S aint-Flour XVIII—1988 , 1–11 2, Lecture Notes in Math., 14 2 7, Springer , Berlin, 199 0. [2] M. Anderson & R. Scho en, Positive harmonic functions on complete manifolds of negative curv ature, Ann. of Math. 121 (198 5), no. 3, 42 9–461 . HARMONI C MEASURES VERSUS QUASICONFORMAL MEAS URES F OR HY PERBOLIC GR OUPS 51 [3] W. Ballmann & F. Le dr appier, Disc r etization of p os itive har mo nic functions on Riemannia n manifolds and Martin b oundary , A ctes de la T able R onde de G´ eom´ etrie Diﬀ ´ er ent iel le (Luminy, 199 2) , S ´ emin. Congr., vol. 1, So c. Math. F r ance, Paris, 1996 , pp. 7 7–92 . [4] F. Be n Nasr , I. Bhouri & Y. Heur tea ux, The v alidit y of the multifractal for malism: results and examples, A dv. Math. 165 (200 2), no. 2, 26 4 –284 . [5] G. Besso n, G. Co ur tois & S. Gallot, E n tropies et rigidit´ es des espac e s lo c a lement sym´ etriques de courbure strictement n´ ega tive, Ge om. F unct. A nal. 5 (1995), no. 5, 73 1 –799 . [6] M. Bjorklund, Ce n tral limit theorems for Gr o mov hyper bo lic g roups, pre print (2 0 08). [7] S. Blach ` ere & S. Br oﬀerio, Internal diﬀusion limited a ggreg a tion on dis crete groups having exponential growth, Pr ob ab. The ory R elate d Fields 13 7 (20 07), no. 3 -4, 32 3–34 3. [8] S. Blach ` ere, P . Ha ¨ ıssinsky & P . Mathieu, Asymptotic ent ropy and Gree n speed for random walks o n coun t- able gro ups, Ann. Pr ob ab. 36 (2008), no. 3, 1134– 1152 . [9] M. Bonk & B. Kleiner , Rigidit y for qua si-M¨ obius gro up actions, J . Diﬀer ential Ge om. 61 (2002), no. 1, 81–10 6. [10] M. Bonk & O. Schramm, Embeddings of Gr omov hyper bo lic spaces, Ge om. F u nct. A nal. 1 0 (2000), no. 2, 266 –306. [11] M. Bourdon & H. Pa jot, Q uasi-confor mal geometry and hyperb olic geometry , Rigi dity in dynamics and ge ometry (Cambridge, 2000) , 1 –17, Spr inger, Berlin, 2002 . [12] B. Bowditc h, Cut p oints a nd cano nical splittings of hyper b o lic g r oups, A cta Math. 18 0 (19 98), no. 2, 145–1 86. [13] R. Bow en & C. Series, Mar ko v maps asso cia ted with F uchsian groups, In s t. Haut es tu des Sci. Publ. Math. 50 (1979 ), 153– 170. [14] C. Connell & R. Muchnik, Harmonicity of quas ic onformal measures and Poisson b oundar ies of hyperb olic spaces, Ge om. F unct . Anal. 17 (2007 ), 707– 769. [15] M. Co orna e rt, Mesure s de Patterson-Sulliv an sur le bo rd d’un e space hyperb olique au sens de Gromo v, Paciﬁc J . Math. 1 59 (1 993), no. 2, 241 –270. [16] B. Deroin, V. Kleptsyn & A. Nav as, On the ques tio n o f ergo dicity for minimal gro up actions on the circle, preprint (2008). [17] Y. Derriennic, Sur le th´ e or` eme er go dique sous -additif, C. R. A c ad. Sci. Paris S´ er. A-B , 2 81 (19 75), no. 22, Aii, A985– A988. [18] E. Dynkin, The b o undary theory o f Markov pro ce s ses (discrete case), Usp ehi Mat. N auk 2 4 (196 9 ), no. 2, 3–42. [19] Y. Guiv arc’h, Sur la loi des grands nom bres et le rayon sp ectral d’une marche al´ eatoir e, Confer enc e on R andom Walks (Kle eb ach, 1979) , Ast´ er isque, vol. 74 , So c. Math. F r ance, Paris, 1980 , pp. 47–9 8, 3. [20] Y. Guiv a rc’h & Y. Le Jan, Sur l’enroulement du ﬂot go dsique. C. R. A c ad. Sci. Pari s Ser. I Math. 311 (1990), no. 10, 645– 648. [21] Y. Guiv arc’h & Y. Le Ja n, Asymptotic winding of the geo de s ic ﬂow on mo dular surfac e s a nd con tin uous fractions, Ann. Sc. Ec. Norm. sup. ( 4) 2 6 (19 93), no. 1 , 23 –50. [22] E. Ghys and P . de la Harp e (´ editeurs), Sur les gr oup es hyp erb oliques d’apr ` es Mikha el Gr omov , Pro gress in Mathematics, 83. Birk h¨ a user Boston, Inc., Bos ton, MA, 19 90. [23] J. Heinonen, L e ctu r es on analysis on metric sp ac es , Universitext. Springer- V erla g, New Y or k, 2 001. [24] G.A. Hunt, Markoﬀ chains and Mar tin b oundaries , Il linois J. Math. 4 (1960 ), 31 3 –340 . [25] V. Ka imanovic h, Brownian motion a nd har monic functions on cov ering manifolds. An entropic approa ch, Soviet Math. Dokl. 33 (19 86), 81 2–816 . [26] V. Kaimanovic h, Inv aria nt measur es of the geo desic ﬂow and measures at inﬁnity on negatively curved manifolds. Hyp er bo lic b ehaviour of dynamical systems, A nn. Inst. H. Poinc ar´ e Phys. Th ´ eor. 5 3 (19 90), no. 4, 361– 393. [27] V. Kaima novic h, Discre tiza tion of b ounded harmo nic functions on Riema nnian manifolds and entrop y , Potential the ory (Nagoya, 1990) , 21 3–22 3 , de Gruyter , Berlin, 1992 . [28] V. K aimanovic h, Ergo dicity of harmonic inv a riant measur es for the geo des ic ﬂow on hyperb olic s pa ces, J. R eine Angew. Math. 4 55 (1 9 94), 57 –103 . [29] V. Ka ima novic h, H ausdor ﬀ dimension of the harmonic measure o n tre e s , Er go dic The ory Dynam. Systems 18 (1998 ), 631– 660. [30] V. Ka ima novic h, The Poisson formula for groups with hyperb olic prop erties, Ann. of Math. 152 (2000), 659–6 92. [31] I. Kap ovich & N. Benakli, Bo undaries of hyperb olic g roups, Combinatorial and ge ometric gr oup the ory (New Y ork, 2000/Hob oken, NJ, 2001 ) , 39–9 3, Co nt emp. Math., 296 , Amer. Math. So c ., Providence, RI, 2002. 52 S ´ EBASTIEN BLACH ` ERE, PETER HA ¨ ISSINS KY & PIERR E MA THI EU [32] A. Ka r lsson & F. Ledra ppier, Pro pri´ et´ e de Lio uville et vitesse de fuite du mouvemen t br ownien, C. R. A c ad. Sci. Paris, S´ er. I 34 4 (20 07), 685– 690. [33] J. F. C. Kingman, The ergo dic theor y o f subadditive sto chastic pro cesses, J. Ro y. S tatist. So c. Ser. B 30 (1968), 499 –510. [34] V. Le Prince, Dimensional prop erties o f the harmonic measur e for a ra ndom walk o n a hyperb o lic group, T ra ns. A mer. Math. So c. 3 59 (200 7), 2 8 81–2 898. [35] V. Le Prince, A relation b etw een dimension of the harmonic measure, ent ropy and drift fo r a random w alk on a hyperb olic space, Ele ct. Comm. Pr ob a. 13 (2 0 08), 45–5 3. [36] F. Ledra ppier, Ergo dic prop erties of Brownian motion on cov ers of compa ct negatively-curv e manifolds , Bol. So c. Br asil. Mat. 19 (1988), no . 1, 115– 1 40. [37] F. Ledra ppier, Harmo nic measures and Bow en-Margulis meas ures, Isr ael J. Math. 71 (199 0), no . 3, 2 75– 287. [38] F. Ledra ppier, Some a symptotic prop er ties of ra ndom walks on free g roups, T opics in pr ob abili ty and Lie gr oups: b oun dary the ory , 117 –152, CRM P ro c. Lectur e Notes, 28, Amer. Math. So c., Providence, RI, 2001. [39] R. Lyons, Equiv alence o f bo undary measures on cov ering trees of ﬁnite graphs, Er go dic The ory Dynam. Systems 14 (1994 ), no. 3, 575 – 597. [40] T. Lyons & D. Sulliv an, F unction theor y , r andom paths and covering spa ces, J. Diﬀer ential Ge om. 19 (1984), no. 2, 299–3 23. [41] J. Maires se & F. Math´ eus, Random walks on fr e e pro ducts of cyclic g r oups J. L ond. Math. So c. ( 2) 75 (2007), no. 1, 47–66 . [42] P . Ma ttila, Ge ometry of sets and me asur es in Euclide an sp ac es. F r actals and r e ctiﬁabil ity , Cambridge Studies in Adv anced Ma thematics, 4 4, Cambridge Universit y Pres s , Cambridge, 19 95. [43] L. Na ¨ ım, Sur le rˆ o le de la fr onti ` ere de R. S. Mar tin dans la th´ eo rie du p otentiel, Ann. Inst. F ourier 7 (1957), 183 –281. [44] M. Pinsky , Stochastic Riemannian geometry , Pr ob abili stic A nalysis and R elate d T opics , ed. A. T. Bharucha-Reid (Academic Pres s, 197 8 ), pp. 199– 236. [45] J.-J . Prat, ´ Etude a s ymptotique et conv ergence angulair e du mouvemen t brownien sur une v a ri´ et´ e ` a co ur - bure n´ ega tive, C. R. A c ad. Sci. Pari s S ´ er. A-B 280 (1975), no . 22, Aiii, A1539–A15 42. [46] J. R. Stallings, On torsio n-free groups with inﬁnitely many ends, Ann. of Math. (2) 88 (196 8), 3 12–3 34. [47] J. V¨ ais¨ al¨ a, Gro mov hyperb o lic spaces, Ex p o. Math. 23 (2005 ), no. 3, 187– 231. [48] A. M. V ershik , Dynamic theor y o f gr owth in groups: entropy , b oundar ies, examples, U sp ekhi Mat. Nauk 55 (2000 ), no. 4(334), 59– 128, (T ra ns lation in Russian Math. Surveys 55 (2 000), no. 4, 6 67–7 33). [49] W. W o ess, R andom wal ks on inﬁn ite gr aphs and gr oups , Cam bridge T racts in Ma thematics, 138. Cam- bridge Universit y Pr ess, Cambridge, 200 0. [50] L.-S. Y o ung, Dimension, ent ropy and Lyapuno v expo nent s, Er go dic The ory Dynam. Systems 2 (198 2), 109–1 24. Eurandom, P.O. Box 513, 5600 MB Eindhove n, The N etherlands, LA TP/CMI, Universit ´ e de Pro vence, 39 r ue Fr ´ ed ´ eric Joliot-Curie, 13453 Marseil l e Cedex 13, France LA TP/CMI, U niversit ´ e d e Pro vence, 39 rue Fr ´ ed ´ eric Jo l iot-Curie, 13453 Marseille Cedex 13, France

Harmonic measures versus quasiconformal measures for hyperbolic groups

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment