The action of a nilpotent group on its horofunction boundary has finite orbits

THE A CTION OF A NILPOTENT GR OUP ON ITS HOROFUNCTI ON BOUND AR Y HAS FINITE ORBITS CORMAC W ALSH Abstract. W e study t he action o f a nilp oten t group G with finite gene rating set S on its horofunction b oundary . W e show that there is one finite orbit associated to each facet of the p ol ytope obtained by pro j ecting S in to the torsion-free component of th e abelianisation of G . W e also prov e that these are the only finite orbits of Busemann p oints. T o finish off, we examine in detail th e Hei s en berg group with its usual generato rs. 1. Introduction Given a g r oup acting b y isometries on a space, one often finds it useful to study the induced action on some b oundary of the space, for example the h yper b o lic bo undary o f a Gromov hyper bo lic s pace or the ideal boundary of a CA T(0) space. The hop e is that the action on the b oundary is simpler than the action on the space itself. Of course, we may endow any finitely-generated g roup with its word metric, and then the gro up acts isometrica lly on itself by left transla tions. One can then study the induced action on the horofunction b o undary , which exists for any metric space. The case of Z d with an arbitrary finite generating set w as in v estigated b y Rief- fel [6] in connectio n with his work on qua n tum metric spa c es. Es sential to his work was that the action on the horob oundary is amenable and that there are s ufficiently many finite orbits. In [4], Develin studied in grea ter deta il the horo b ounda ry of Z d and the a c tio n of Z d on it. In this pap er, we study the case of a finitely- generated nilp otent group. W e find that, just as in the case of Z d , there are alwa ys finite o rbits, whatev er the generating se t. T o state our re s ults, we need the follo wing c o nstruction. Recall that the quotien t G/ [ G, G ] o f a gro up G b y its commutator subgr o up is an abelian group, called the ab elianisation . If G is finitely generated, then so is the ab elianisation, which can therefore b e written as a dir e c t pro duct Z 0 × Z N , where Z 0 is a finite a b elia n g roup (the torsion s ubgroup) and N ∈ N . So we can define a pro jection map φ assigning to every e le men t of G its equiv alence cla ss in ( G/ [ G, G ]) / Z 0 . W e consider Z N to Date : Nov ember 12, 2018. 2000 Mathematics Subje ct Classific ation. Pri mary 20F65; 20 F18. Key wor ds and phr ases. group action, horoball, max-plus a lgebra, metric boundary , Busemann function. This work was partially supp orted b y the join t RFBR-CNRS gran t n um ber 05-01-02807. 1 2 CORMA C W ALSH be embedded in R N . Let P := con v ( φ ( S )) b e the conv ex hull of the image of the generating se t S under the map φ . W e will a ssume for simplicity that S is s ymmetric, that is, S = S − 1 , s ince this makes the word length distance a metric. The r esults sho uld also b e true, how ever, in the more g eneral case . A pa rticularly interesting subs e t of the ho rofunction b oundar y is its s et o f Buse- mann p oints. These are clo sely rela ted to the g eo desic paths. In gener al, all geo- desic paths conv erge to Busemann p oints [6 ]. Moreover, when the m etric is integer v alued, as in the pr esent setting, the Busemann p oints are precisely the limits o f the g e o desic paths. Recall that a fa cet of a p olytop e is a pro per face of maximal dimensio n, in o ther words, of co-dimension 1. Theorem 1.1. L et G b e a n ilp otent gr oup with finite symmetric gener ating set S and c onsider t he action of G on its hor ofunction b oundary c oming fr om the wor d length metric asso ciate d to S . Then, t her e is a n atur al one-t o-one c orr esp ondenc e b etwe en the fi nite orbits of Busemann p oints and the fac ets of P . When G = Z d , the map φ is just the identit y map. In this ca se, it was shown in [4] that all b oundary p oints ar e Busemann and that there is a one-to-o ne c o r- resp ondence b etw een the or bits and the prop er faces of P , with the finite or bits corres p onding to the facets. W e will give an example later of a non-abelia n nilpo- ten t g roup where this s tr onger cor resp ondence brea k s down. W e also hav e more precis e results for the discrete Heisenberg gr o up, which has the following presentation: H 3 := h a, b | [[ a, b ] , a ] = [[ a, b ] , b ] = e i . W e use the word le ngth metric coming from the usual set of generator s S := { a, b, a − 1 , b − 1 } . F or this group with these g enerators , an exact for m ula for the word le ng th metric is k nown. W e show directly that there a re four finite or bits of Busemann points, each consisting of a single horofunction. Thes e horofunctions ar e the limits o f the four g eo desic words a ǫ a b ǫ b a ǫ a b ǫ b · · · , with ǫ a = ± 1 and ǫ b = ± 1. W e also show that every o ther Busemann p oint is the limit of a geo des ic path of the form t → c m b n a ± t or t → c m ± lt b ± t a l , wher e c := [ a, b ]. Using the formula for the metric, a ll these Busemann p oints can b e ca lculated explicitly; s ee Theo rems 5.3 and 5.4. 2. The Horofunction Boundar y The horofunction b oundary of a prop er metric spa ce ( X , d ) is defined as follows. One as s igns to ea ch p oint z ∈ X the function ψ z : X → R , ψ z ( x ) := d ( x, z ) − d ( b, z ) , where b is some base-p oint. The ma p ψ : X → C ( X ) , z 7→ ψ z defines a con- tin uous injection of X into C ( X ), the space of con tin uous rea l-v alued functions on X endow ed with the top olog y o f unifor m co nvergence on compact sets. The horofunction b oundary is defined to b e X ( ∞ ) := cl { ψ z | z ∈ X }\{ ψ z | z ∈ X } , AC TION OF A NILPOTENT GROUP ON ITS HOR OFUNCTION BOUNDAR Y 3 and its elemen ts are called ho r ofunctions. This definition seems to be due to Gromov [5]. F or mo re informa tion, see [2], [6], and [1]. The action o f the iso metry group Isom( X, d ) o f X extends contin uo usly to a n action by homeomor phisms on the horofunction compac tification. O ne ca n check that g · ξ ( x ) = ξ ( g − 1 · x ) − ξ ( g − 1 · b ) , for any isometry g , p oint x , and horofunction ξ . The metric spaces in which we will b e interested in this pap er are the finitely- generated nilp otent g roups with their word length metrics. Recall that t he Ca yley graph of a group G with symmetric gener a ting set S has a vertex for each element of G and a n edge b etw een tw o vertices u and v if u − 1 v ∈ S . The w ord leng th distance d ( u, v ) be t ween u and v in G is the leng th of the s ho rtest path b etw een the corres p onding vertices in the Cayley gr aph. By constructio n, it is a left inv ar iant metric. W e choose the identit y element e as the base- p oint. A path is a sequence of gro up elemen ts. W e say tha t a pa th is ge o desic if it is an isometry from N to G . It is known that geo desic paths conv erge to p oints in the horo function b oundary . In the pres en t setting, since the metric is integer v alued, we may define a Busemann p oint as the limit of a geodes ic. See [6] and [1] for the definition in more gener al spaces. F or any finite w ord w with letters in S , we deno te b y w the elemen t of G obtained by m ultiplying the letters of w , a nd we say that w r epr esents w . W e denote by | w | the length of w , that is, its num b er of letters. W e say that a finite word w is a ge o desic wor d if | w | = d ( e, w ), and that a n infinite word is geo desic if each of its finite prefixes is. Obviously , ther e is a one-to- one co rresp ondence betw een geo desic paths a nd pairs ( g , w ), where g is in G ( the starting point of the path) and w is an infinite geodesic w ord. The steps o f a geo desic path are the letters of the cor resp onding geo desic word. Contin uing to use termino logy of [4], we say that the dir e ction of a geo desic path is the set of letters t hat o ccur infinitely often as steps. W e s tart o ff w ith an easy prop ositio n telling us when tw o geo de s ics co nv er ge to the same Busemann p oint. Note that the proof relies o n the fact that the horofunctions ar e integer v alued. Prop ositio n 2.1. L et γ 1 and γ 2 b e two ge o desics i n a finitely-gener ate d gr oup. The fol lowing ar e e quivalent: (i) γ 1 and γ 2 c onver ge to the same Busemann p oint; (ii) for al l N ∈ N , ther e is a ge o desic agr e eing with γ 1 up t o time N that has a p oint in c ommon with γ 2 and eventual ly agr e es with γ 1 again; (iii) ther e is a ge o desic having infinitely many p oints in c ommon with b oth γ 1 and γ 2 . Mor e over, the ab ove st atements hold if (iv) γ 1 and γ 2 have a p oint in c ommon, and, for al l N ∈ N , ther e is a ge o desic agr e eing with γ 1 up to t ime N with the same limit as γ 2 , and a ge o desic agr e eing with γ 2 up t o time N with t he same limit as γ 1 Pr o of. The implicatio ns (ii) ⇒ (iii) and (iii) ⇒ (i) are obvious. 4 CORMA C W ALSH Assume (i) holds, tha t is, γ 1 and γ 2 conv erge to some hor ofunction ξ . Let N ∈ N . W e can find t 2 large enoug h that ξ ( γ 2 (0)) − ξ ( γ 1 ( N )) = d ( γ 2 (0) , γ 2 ( t 2 )) − d ( γ 1 ( N ) , γ 2 ( t 2 )) . W e can then find t 1 large enoug h that ξ ( γ 2 ( t 2 )) − ξ ( γ 1 ( N )) = d ( γ 2 ( t 2 ) , γ 1 ( t 1 )) − d ( γ 1 ( N ) , γ 1 ( t 1 )) . Since γ 2 is a geo des ic, ξ ( γ 2 (0)) = d ( γ 2 (0) , γ 2 ( t 2 )) + ξ ( γ 2 ( t 2 )) . It follows that d ( γ 1 ( N ) , γ 1 ( t 1 )) = d ( γ 1 ( N ) , γ 2 ( t 2 )) + d ( γ 2 ( t 2 ) , γ 1 ( t 1 )) . So there exists a geo desic agr eeing with γ 1 up to time N , passing through γ 2 ( t 2 ), and ag r eeing with γ 1 after time t 1 . This establishes (ii). Now assume that (iv) holds. So there is a geo desic γ 3 agreeing with γ 1 up to any given time N with the same limit a s γ 2 . Applying the eq uiv alence of (i) and (ii) just prov ed, we see ther e is a geo desic γ 4 agreeing with γ 1 up to time N and agreeing with γ 2 after some time M . A symmetrical a rgument shows that there is a geo desic γ 5 agreeing with γ 2 up to time M and event ually ag reeing with γ 1 . It is easy to pr ov e tha t the path a g reeing with γ 4 up to time M and therea fter agreeing with γ 5 is a geo des ic satisfying the requirements of (ii).  3. Preliminaries on Nilpotent G r oups W e use [ a, b ] to deno te the co mm utator a − 1 b − 1 ab of tw o elements of a group. Also, for an y t wo s ubgroups A and B , w e deno te by [ A, B ] the subgro up g enerated by all [ a, b ] with a ∈ A and b ∈ B . F or a ny group G , the subgro up [ G, G ] is called the c ommutator sub gr oup . It is alwa ys normal. Recall tha t a group G is nilp otent if Γ n is triv ial for some n ∈ N , where Γ i ; i ∈ N is the lower c entr al series defined by Γ 1 := G a nd Γ i +1 := [Γ i , G ] for i ∈ N . W e will need the fo llowing tw o lemmas concerning finite-index subgroups of nilpo ten t g roups. They have similar pro ofs, which rely on the ident ities [ x, y z ] = [ x, z ][ x, y ][[ x, y ] , z ] and (1) [ xy , z ] = [ x, z ][[ x, z ] , y ][ y , z ] . (2) Lemma 3.1. L et H b e a fin ite index sub gr oup of a finitely gener ate d nilp otent gr oup G . Th en, [ H , H ] has fi nite index in [ G, G ] . Pr o of. W e use an inductive ar gument. The induction hypothesis is that there is a finite set R i ⊂ G such that any g ∈ [ G, G ] can b e expr essed as g = h ′ r ′ mo d Γ i , with h ′ ∈ [ H , H ] a nd r ′ ∈ R i . The ca se when i = 2 is trivia l. Now assume the hypo thesis is tr ue for s ome i ≥ 2. The a ssumption of the lemma implies that, for each generato r s ∈ S , there is some n s ≥ 1 and h s ∈ H such that s n s = h s . Let C i denote the set o f simple commutators o f weigh t i of elements of the genera ting set S . In other words, C 1 := S and C j +1 :=  [ c, s ] | c ∈ C j and s ∈ S  , for j ≥ 1 . AC TION OF A NILPOTENT GROUP ON ITS HOR OFUNCTION BOUNDAR Y 5 Let c := [ . . . [ s 1 , . . . ] , s i ] b e an ar bitrary element of C i . Using the iden tities (1) and (2), one can show that c n s 1 ··· n s i = [ . . . [ s n s 1 1 , . . . ] , s n s i i ] mo d Γ i +1 = [ . . . [ h s 1 , . . . ] , h s i ] mo d Γ i +1 . So, letting p c := n s 1 · · · n s i , we see that c p c is in [ H , H ], mo dulo Γ i +1 . As Γ i / Γ i +1 is ab elian and ge nerated by the elements o f C i mo d Γ i +1 we can write any d ∈ Γ i in the form d = Y c ∈ C i c m c mo d Γ i +1 , where the product is taken in any fix e d o rder and m c ∈ Z for all c ∈ C i . Therefore, from what we hav e just proved, d = h Y c ∈ C i c q c mo d Γ i +1 , for so me h ∈ [ H, H ], where q c is the remainder when m c is divide d by p c . Let R be the finite set R := n Y c ∈ C i c q c | 0 ≤ q c < p c for all c ∈ C i o . By the induction h yp othesis, ther e is a finite set R i ⊂ G such that any g ∈ [ G, G ] can b e express ed as g = h ′ r ′ d , with h ′ ∈ [ H , H ], r ′ ∈ R i , and d ∈ Γ i . So g = h ′ dr ′ mo d Γ i +1 . B y the result of the pr evious parag raph, d = hr mo d Γ i +1 for some h ∈ [ H, H ] and r ∈ R . Therefor e, g = h ′ hrr ′ mo d Γ i +1 , and so the inductio n hypothesis is true for i + 1.  Lemma 3.2. L et H b e a sub gr oup of a finitely gener ate d nilp otent gr oup G , and let Θ b e the natu ra l homomorphism fr om G to its ab elianisation G/ [ G, G ] . If Θ( H ) has finite index in Θ( G ) = G/ [ G, G ] , then H h as fin ite index in G . Pr o of. W e use again a n inductive argument. The induction h ypo thesis here is that there is a finite s et R ′ ⊂ G s uch that any g ∈ G can b e expressed as g = h ′ r ′ mo d Γ i , with h ′ ∈ H and r ′ ∈ R ′ . By assumption, this is true for i = 2 . Now a ssume it is true for some i ≥ 2. The assumption of the lemma implies that, for each gener ator s ∈ S , there is s o me n s ≥ 1 and h s ∈ H such that Θ( s n s ) = Θ( h s ). Therefore, for all s ∈ S , there exists g s ∈ [ G, G ] suc h that s n s = h s g s . As in the pro of of the previous lemma, let C i denote the set of simple com- m utators of weight i of elemen ts o f S , and consider an a rbitrary element c := [ . . . [ s 1 , . . . ] , s i ]. Using the identities (1) and (2 ), one can s how as b efore tha t c p c = [ . . . [ s n s 1 1 , . . . ] , s n s i i ] mo d Γ i +1 = [ . . . [ h s 1 , . . . ] , h s i ] mo d Γ i +1 , where p c := n s 1 · · · n s i . 6 CORMA C W ALSH It follows just as befo re tha t any d ∈ Γ i can b e written as d = hr mo d Γ i +1 for so me h ∈ H a nd r ∈ R , whe r e R is the finite set R := n Y c ∈ C i c q c | 0 ≤ q c < p c for all c ∈ C i o . By the induction hypothesis there is a finite set R ′ ⊂ G such that a ny g ∈ G can b e e xpressed a s g = h ′ r ′ d , with h ′ ∈ H , r ′ ∈ R ′ , and d ∈ Γ i . So g = h ′ dr ′ mo d Γ i +1 . But we have seen that d = hr m o d Γ i +1 for some h ∈ H and r ∈ R , and therefor e, g = h ′ hrr ′ mo d Γ i +1 . This co mpletes the induction step.  4. F a cets and Finite Orbits Let G be a nilp otent gro up genera ted by a finite symmetric set S . Recall that φ is the pro jection onto the torsion-free co mpo nen t of the a belia nisation o f G , and that P is the conv ex hull of φ ( S ), considered as a subset of R N . Lemma 4.1. L et V b e a subset of S such that φ ( V ) is c ontaine d in a fac et of P . Then, any wor d with lett ers in V is ge o desic. Pr o of. Let F ⊂ R n be the facet into whic h V is mapped, a nd le t f : R n → R be the linea r functional de fining F , that is, such that f ( x ) = 1 for x ∈ F . So, if z 0 z 1 · · · is a word with le tter s in V , then f ◦ φ ( z 0 · · · z n − 1 ) = n for all n ∈ Z . Let y 0 y 1 · · · be a word with letters in S such that y 0 · · · y m − 1 = z 0 · · · z n − 1 for some n a nd m in Z . Since f ( x ) ≤ 1 for x ∈ P , we have m ≥ f ◦ φ ( y 0 · · · y m − 1 ) = n . It follows that z 0 z 1 · · · is a g eo desic word.  Lemma 4. 2. L et H b e a nilp otent gr oup gener ate d as a gr oup by a finite set V . Then, for any g ∈ H , we c an write g x = y , wher e x and y ar e wor ds with letters in V . Pr o of. W e will show the r esult is true for each of the gr oups H / Γ i ; i ≥ 2, using induction. It is clea rly true for the gr oup H / Γ 2 = H / [ H , H ], since this group is ab elian. Now assume the result is true for H / Γ i for so me i ≥ 2. Let C j denote the set of simple commutators o f weigh t j of elements of V . In other words, C 1 := V a nd C j +1 :=  [ c, s ] | c ∈ C j and s ∈ V  , for j ≥ 1 . F rom the induction hypo thesis, for every c ∈ C i − 1 , there exis ts words x c and y c with letters in V suc h that c x c = y c mo d Γ i . F or each c ∈ C i − 1 , let g c ∈ Γ i be such that cx c = y c g c . (3) Consider the word w := Y c ∈ C i − 1 s ∈ V ( y c s n ( c,s ) x c )( x c s m ( c,s ) y c ) , where the pro duct is taken in any fixe d orde r , and n ( c, s ) and m ( c, s ) are non- negative in tegers for all c ∈ C i − 1 and s ∈ V . Clear ly , the letters of w are in V . AC TION OF A NILPOTENT GROUP ON ITS HOR OFUNCTION BOUNDAR Y 7 Using (3) and the fact that [ c, h ] and g c are central in H / Γ i +1 for every c ∈ C i − 1 and h ∈ H , we get y c s n ( c,s ) x c = cx c g − 1 c s n ( c,s ) c − 1 y c g c , mo d Γ i +1 = x c [ c, x c ] s n ( c,s ) [ c, s ] n ( c,s ) y c , mo d Γ i +1 . Similarly , x c s m ( c,s ) y c = y c [ x c , c ] s m ( c,s ) [ s, c ] m ( c,s ) x c , mo d Γ i +1 . So w := Y c ∈ C i − 1 s ∈ V [ c, s ] n ( c,s ) − m ( c,s ) Y c ∈ C i − 1 s ∈ V ( x c s n ( c,s ) y c )( y c s m ( c,s ) x c ) , mo d Γ i +1 . Since the gr oup Γ i / Γ i +1 is gener a ted b y C i mo d Γ i +1 , and is in the cen ter of H/ Γ i +1 , any h ∈ Γ i can be written (mod Γ i +1 ) as the first fa c to r in the righ t- hand-side of the ab ov e equation, with a n appropriate choice of n and m . The second factor can be represented by a w ord z with letters in V . So h z = w mo d Γ i +1 . Now let g ∈ H . By the induction h yp othesis, g x = y mo d Γ i for some w o rds x and y with le tter s in V . In other words, g xh = y for some h ∈ Γ i . As w e hav e just seen, h z = w mod Γ i +1 for so me words z and w w ith letters in V . So g xw = yz mo d Γ i +1 . This completes the induction step.  W e are now ready to co nstruct geo des ic pa ths leading to the B usemann p oints lying in finite orbits. Let F be a facet o f P , and let V b e the set of g enerators mapp ed into F b y φ . W e denote by h V i the subgro up of G g e nerated as a g roup by the elements of V . O f cour se, h V i is also nilp otent. So, by Lemma 4.2, for ea ch g ∈ h V i , there exist words x g and y g with letters in V suc h that g x g = y g . Let C i be the set of simple commutators of weigh t i of elements of V . T a ke a finite word w F with letters in V such that x c and y c app ear as sub w ords, for e very c ∈ S i C i . The infinite word w F w F · · · is geo desic, by Lemma 4.1. W e define Λ F : N → G to b e the geo desic path starting a t the identit y , having steps given by this word. Since Λ F is geo desic, it conv erges to a Bus e mann p oint of the horob oundar y , which we denote by ξ F . Lemma 4.3. L et F b e a fac et of P , and let V b e the s et of gener ators mapp e d into F by φ . Then, for any g ∈ h V i , t her e is an n ∈ N such that g w n F c an b e written as a pr o duct of elements of V . Pr o of. Let Γ i ( h V i ) deno te the decending ce ntral ser ies o f h V i , and let C i denote the se t of simple commutators o f weigh t i of elements of V . W e consider the g roups h V i / Γ i ( h V i ); i ≥ 1 and use induction. The ca se when i = 1 is trivia l. Now ass ume that g w n F can be written (mo d Γ i ( h V i )) in the desired form. Tha t is to say , g w n F = v c for so me word v with letter s in V and some c ∈ Γ i ( h V i ). W e know that c can b e written as a product o f simple comm utators a nd their inverses: c = c ǫ 1 1 · · · c ǫ m m , mo d Γ i +1 ( h V i ) , 8 CORMA C W ALSH where m ∈ N , and c j ∈ C i and ǫ j = ± 1 for all j ∈ { 1 , . . . , m } . Recall that Γ i ( h V i ) / Γ i +1 ( h V i ) is in the center of h V i / Γ i +1 ( h V i ). Supp ose t hat ǫ j = 1 for some j . Reca ll that w F has subw ords x c j and y c j satisfying c j x c j = y c j . So by replacing the subw ord x c j in w F by y c j , we obta in a word with letters in V that represents c j w F mo d Γ i +1 ( h V i ). Similarly , for an y j for which ǫ j = − 1 , we replace y c j by x c j to obtain a word with letters in V represe nting c − 1 j w F mo d Γ i +1 ( h V i ). W e concatenate all these words to get a w ord u with letters in V representing cw m F mo d Γ i +1 ( h V i ). So v u represen ts g w n + m F mo d Γ i +1 ( h V i ). This concludes the induction s tep.  Lemma 4.4. L et F b e a fac et of P , and let V b e the s et of gener ators mapp e d into F by φ . Th en, the hor ofunction ξ F is invariant under t he sub gr oup h V i . Pr o of. Let g ∈ h V i and n ∈ N . Since g [ g , w n F ] ∈ h V i , we ca n find, by Le mma 4 .3, m ∈ N such that g [ g , w n F ] w m F is represented b y some word v with letter s in V . Similarly , w e can find l ∈ N suc h that [ w n + m F , g ] g − 1 w l F is represen ted by a w o rd v ′ with letters in V . Consider the path γ star ting at the identit y with steps given by the word w n F v v ′ w F w F · · · . By Lemma 4.1, it is a geo desic. It ag rees with the geo desic path Λ F up to time n | w F | . But w n F v = g w n + m F , a nd so γ has a po in t in common with the path g Λ F at time | w n F v | . W e als o hav e that w n F v v ′ = w n + m + l F . Therefore, γ a g rees with Λ F after time | w n F v v ′ | . W e now use Prop os ition 2.1 to deduce that Λ F and g Λ F conv erge to the s ame limit ξ F .  Theorem 4.5. L et F b e a fac et of P , and let V b e the set of gener ators m app e d into F by φ . Th en, the st abiliser of ξ F is h V i . Pr o of. W e hav e already see n in Lemma 4.4 tha t h V i sta bilis es ξ F . Let g ∈ G b e such that g ξ F = ξ F . Cons ider the ge o desics Λ F and g Λ F . Since they hav e the sa me limit ξ F , there exis ts a g e o desic γ having infinitely many p o int s in common with each, by Pr op osition 2.1. So ther e exist t 0 , t 1 , a nd t 2 in N with t 0 ≤ t 1 ≤ t 2 , such that γ ( t 0 ) = Λ F ( s 0 ), γ ( t 1 ) = g Λ F ( s 1 ), and γ ( t 2 ) = Λ F ( s 2 ) for some s 0 , s 1 , and s 2 in N . Let f : R n → R b e the linear functional defining the facet F , so that f ( x ) = 1 for x ∈ F and f ( x ) < 1 for x ∈ P \ F . Since the steps of Λ F lie in V , we hav e f ◦ φ (Λ F ( s 2 )) − f ◦ φ (Λ F ( s 0 )) = s 2 − s 0 . But since γ is a ge o desic, s 2 − s 0 = t 2 − t 0 . W e conclude that f ◦ φ ( γ ( t 2 )) − f ◦ φ ( γ ( t 0 )) = t 2 − t 0 . Using the fact that f ◦ φ is a homomorphism a nd that f ◦ φ ( x ) ≤ 1 for x ∈ S , we get that ev ery step x of γ b etw een t 0 and t 2 satisfies f ◦ φ ( x ) = 1, that is, they all lie in V . Consider the finite path that star ts at the iden tity , is identical to Λ F up to time s 0 , follows γ from γ ( t 0 ) = Λ F ( s 0 ) to γ ( t 1 ) = g Λ F ( s 1 ), a nd then follows g Λ F backw ards along to g . The steps o f this path lie e ntirely within V ∪ V − 1 . W e deduce that g ∈ h V i .  AC TION OF A NILPOTENT GROUP ON ITS HOR OFUNCTION BOUNDAR Y 9 Corollary 4 .6. F or every fac et F of P , the Busemann p oint ξ F lies in a finite orbit. Pr o of. Recall tha t Z 0 is the torsion subgroup of the ab elianisation of G . Cho ose a subset R of G containing one representative of each equiv a lence class in Z 0 . Since F is a fac e t o f P , the set of vectors φ ( V ) generates a finite ind ex subg roup of Z N . So there is a finite subset R ′ of G s uc h that, for any g ∈ G , ther e exists r ′ ∈ R ′ and v ∈ h V i sa tisfying φ ( g ) = φ ( v ) φ ( r ′ ). So Θ( g ) = Θ( v r ′ ) mo d Z 0 , where Θ is the natural homomo rphism from G to its ab elianisation G/ [ G, G ]. This is equiv ale n t to Θ ( g ) = Θ ( v )Θ( rr ′ ) for some r ∈ R . W e conclude from this that Θ ( h V i ) has finite index in Θ( G ). So, by Lemma 3.2, h V i has finite index in G . But, by the theor em, h V i is exactly th e stabiliser of ξ F , and so this Busemann p oint has a finite or bit.  W e hav e just shown that to every facet of P there co rresp onds a finite orbit of Busemann p oints. T o pr ov e the co rresp ondence in the opp os ite dir e c tion, we will need the following le mma. Note that its hyp o thesis that the o rbit is finite is no t needed in the ab elian setting [4]. Lemma 4.7. Supp ose the limit of a ge o desic p ath γ li es in a finite orbit. Then, the pr oje ction map φ maps the dir e ction of γ into some fac et of P . Pr o of. Let u 0 u 1 · · · be the geo desic word giving the s teps of γ . F or each n ∈ N , let γ n be the geo des ic starting at the iden tit y and having steps given by the word u n u n +1 · · · . The limit of ea ch γ n is in the same orbit as that of γ , which we hav e assumed finite. Therefore, there is some subs e quence of geo desics γ n i all of whose limits a re equal. Ca ll the common limit ξ . F or each i ∈ N , define the finite word w i := u n i · · · u n i +1 − 1 , a nd let W := { w i | i ∈ N } b e the set of all these words. Fix i ∈ N . Since γ n i is even tually ju st a copy of w i γ n i +1 shifted in time, these tw o geo desics have the same limit. But b oth γ n i and γ n i +1 conv erge to ξ . Therefore, w i represents a group element in th e stabiliser of ξ . Now cons ider an infinite concatenation z := z 0 z 1 · · · z m · · · o f w ords z m ; m ∈ N , each in W . Given any m ∈ N , the word z can b e written z = z 0 · · · z m − 1 u n i · · · u n i +1 − 1 z m +1 · · · bec ause z m = w i for some i ∈ N . Since p := z 0 · · · z m − 1 is in the stabiliser of ξ , we ha ve ξ ( pg ) = ξ ( p ) + ξ ( g ) for all g ∈ G . So ξ ( z 0 · · · z m − 1 u n i · · · u n i + l − 1 ) = ξ ( z 0 · · · z m − 1 ) + ξ ( u n i · · · u n i + l − 1 ) , for l ∈ { 0 , . . . , n i +1 − n i } . But since γ n i is a ge o desic converging to ξ , w e hav e ξ ( u n i · · · u n i + l − 1 ) = − l . Since this rea soning is v alid for any m ∈ N , we conclude that ξ , ev alua ted along the path λ , decreases at every step, where λ is the path starting a t the identit y with steps given by the word z . It follows from this that λ is a g eo desic. W e hav e th us shown that every co ncatenation o f words fr o m W is a geo desic word. T ake no w sufficiently man y words z 0 , . . . , z n − 1 from W that ev ery gener a tor in the direction of γ appe a rs in at least o ne. Let a := φ ( z 0 · · · z n − 1 ), and let b b e the intersection o f the line 0 a in R N with the boundary of P . O f c ourse, b has 10 CORMA C W ALSH rational co o rdinates and lies in some facet F of P . So, by Lemma 7 of [4], b may be expressed as an affine combination o f the vertices of F , with non-negative rationa l co efficients. Since a is a ra tio nal m ultiple of b , we may express a in the same wa y . Multiplying thr o ugh by the common denomina tor, we can find a p ositive integer l as large as w e wis h, such that l a is a line a r com bination of the vertices of F with non-nega tive in teger co efficients. So there e xists a word x with letters in φ − 1 ( F ) ∩ S such that φ ( x ) = l a . Recall that Z 0 is the torsion subgroup o f the abelianis a tion G/ [ G, G ]. Let T ⊂ G be such that ther e is exa c tly one element o f T in each o f the equiv alence classes in Z 0 . Of course, | T | = | Z 0 | < ∞ . W e can now write ( z 0 · · · z n − 1 ) l = xtc , with t ∈ T and c ∈ [ G, G ]. W e hav e a s sumed that the stabiliser o f ξ , which we no w call U , has fi nite index in G . Therefore, b y Le mma 3.1, [ U, U ] has finite index in [ G, G ]. So we can write c = ru , with r in some finite subset R of G , a nd u in [ U, U ]. Because U stabilises ξ , the res triction of ξ to U is a homomo rphism into Z . In particular, ξ ( u − 1 ) = 0. So, for i ∈ N large eno ugh, d ( u − 1 , γ n 0 ( i )) = d ( e, γ n 0 ( i )) = i . Let D := max { d ( e, t ′ r ′ ) | t ′ ∈ T and r ′ ∈ R } . Of cours e, D is finite. W e hav e d ( e, xtr uγ n 0 ( i )) ≤ d ( e, x ) + d ( e, tr ) + d ( e, uγ n 0 ( i )) ≤ | x | + D + i. But | x | = l f ( a ), where f : R N → R is the linear functional s uch that f ( g ) = 1 for all g ∈ F . Also, since ( z 0 · · · z n − 1 ) l w 0 w 1 w 2 · · · is a g eo desic word, d ( e, ( z 0 · · · z n − 1 ) l γ n 0 ( i )) = l | z 0 · · · z n − 1 | + i. Therefore lf ( a ) + D ≥ l | z 0 · · · z n − 1 | . Since l ca n be made arbitra rily large, f ( a ) ≥ | z 0 · · · z n − 1 | . W e may now apply the fact that f ◦ φ ( s ) ≤ 1 for every s ∈ S to deduce that f ◦ φ ( g ) = 1 f or every letter g in z 0 · · · z n − 1 , which includes ev ery letter in the direction of γ . Therefore, all these letters are mapp ed by φ to F .  Theorem 4.8. Every fi nite orbit of Busemann p oints c an b e written { g ξ F | g ∈ G } for some fac et F of P . Pr o of. Let γ 1 be a geo desic con v erging to a point in a finite o rbit. By Lemm a 4.7, the ima g e of the direction o f γ 1 under φ lies in some fac e t F of P . By , if necessary , premultiplying b y a n appropriate factor and removing an init ial section of the path, we ca n find a g eo desic path γ conv erging to a p oint ξ in the same orbit a s the limit of γ 1 that s tarts a t the identit y and for whic h the image under φ o f every step lies in F . Let the word w F be defined a s a bove. Since ξ is in a finite o rbit, there exists so me m ∈ N with m > 0 such that w m F ξ is equal to ξ . So w mn F ξ = ξ for a ll n ∈ N . Let z 0 z 1 · · · b e t he geodesic word g iving the steps of γ , and let γ 2 be the geo desic path s tarting at the identit y and having steps given by w mn F z 0 z 1 · · · for some large n ∈ N . By Lemma 4.1, γ 2 is a geo desic. Also , γ 2 agrees with Λ F up to time nm | w F | , whic h can b e made as large as one lik es. Finally , γ 2 conv erges to ξ , and so ha s the same limit as γ . AC TION OF A NILPOTENT GROUP ON ITS HOR OFUNCTION BOUNDAR Y 11 W e ha ve prov ed one half of c ondition (iv) of Prop ositio n 2.1. T o prov e the other, ta ke n as large a s you want and let g := z 0 · · · z n . Clearly , g − 1 ∈ h V i , wher e V is the set of gener ators mapp ed into F by φ . Therefore, by Lemma 4.3, ther e is some m ∈ N such that g − 1 w m F can b e repr esented by some word v with letter s in V . B y Lemma 4.1, the path starting at the ident ity and having steps given b y z 0 · · · z n v w F w F · · · is a geo desic. This path agrees with γ up to time n , and it even tua lly agrees with Λ F bec ause z 0 · · · z n v = w m F . It now follows from Prop osition 2 .1 that γ and Λ F hav e the same limit, in other words, that ξ = ξ F .  W e finish this sectio n by s howing how the correspo ndence in the a be lia n setting betw een prop er faces of P and orbits of Busema nn p oints breaks d own for general finitely-genera ted nilp otent gro ups. Example 1. Consider the following gr oup of nilpo tency cla ss 3: G := h a, b | [ a, g ] = [ b, g ] = [ a, h ] = [ b , h ] = e i , where c = [ a, b ], g := [ a, c ], and h := [ b, c ]. An y ele men t of G can be written in the form g i h j c k b l a m , with i , j , k , l , and m in Z . W e may ta ke the pro jectio n map φ to b e φ ( g i h j c k b l a m ) := ( l , m ). The image of the generating set { a, b, a − 1 , b − 1 } is a square. Let F b e the facet defined by the corner s { (1 , 0) , (0 , 1) } . Define the w o rd x := ab . W e will show that, although xxx · · · and Λ F hav e the same dire ctions, their limits differ . Lemma 4.9. L et w b e any wor d with letters in { a, b } s u ch t hat w = g m h n x l for some m, n ∈ Z and l ∈ N . Then, m − n is p ositive unless w = x l . Pr o of. Represent w as a piecewise affine cur ve that starts at the origin and takes a unit step in the x -direction for each letter b , and unit step in the y -direction for e ach letter a . The curv e will ob viously finish at ( l , l ). W rite w in the form g i h j c k b l a l . It is not hard to see that k is the num b er of unit squar es under the curve. After further thoug ht , one a lso sees that j is the sum of the x -co ordinates of the upp er r ight corners of these s quares, and that i is the sum of the y -co or dinates. So the center of gravit y , taking the mass of each unit square to b e concentrated on its upp e r r ight corner , is ( j /k , i/k ). W e now redraw the figure with the axes rotated through a 45 ◦ angle, as in Figur e 1 . W e see that choosing w to minimise i − j is the same a s finding the curv e b etw een (0 , 0) a nd ( l , l ) having ( l + 1) l / 2 unit sq ua res underneath it that minimises the height of the cen ter of mass of these squares. But it is o b vious that there is a unique minimising curve, namely that obtained fro m x l . The conclusion follows.  Prop ositio n 4. 10. The limit η of xxx · · · is differ ent fr om ξ F . Pr o of. Let l ∈ N and let w b e a geo des ic word with letters in { a, b, a − 1 , b − 1 } representing g − 1 x l . By Lemma 4.9, either a − 1 or b − 1 o ccurs a s a letter o f w . Also, the excess of a s ov er a − 1 s in w equals l , as do es the excess o f b s ov er b − 1 s. Therefore, d ( e, g − 1 x l ) = | w | > 2 l . W e conclude tha t η ( g ) = lim l →∞  d ( g , x l ) − d ( e, x l )  > 0 . 12 CORMA C W ALSH b a a (0 , 0) b (8 , 8) (8 , 0) Figure 1. Diagram for the pro of of Lemma 4.9. How ever, { a, b } gener ates G , a nd so ξ F is a fixed p oint b y Theor em 4.5. So ξ F is a ho momorphism from G to Z . In par ticular, ξ F ( g ) = 0. Therefore, η and ξ F differ.  5. The boundar y of the discrete H eisenberg group The discr e te Heisenberg group H 3 is the group of 3 × 3 uppe r triangula r matrices of the form   1 x z 0 1 y 0 0 1   , with x , y , and z in Z . It is the simplest no n-ab elian nilp otent gro up. Let a :=   1 1 0 0 1 0 0 0 1   , b :=   1 0 0 0 1 1 0 0 1   , and c :=   1 0 1 0 1 0 0 0 1   . Observe that ab = bac , that is, c is the commutator of a a nd b , and that th e cen ter of H 3 is the cyc lic gro up generated by c . The following is a pres ent ation of H 3 : H 3 := h a, b | [[ a, b ] , a ] = [[ a, b ] , b ] = e i . All elements of H 3 can b e written in the form c z b y a x , with x, y , z ∈ Z . W e will ca lculate a ll the B usemann p oints of this gro up with the word leng th metric coming from the standa rd gener ating s et S := { a, b , a − 1 , b − 1 } . Our work relies on the following form ula for this metric, found by Blach` ere [3]. AC TION OF A NILPOTENT GROUP ON ITS HOR OFUNCTION BOUNDAR Y 13 Theorem 5.1 ([3]) . L et g := c z b y a x b e an element of H 3 . Its distanc e to the identity with r esp e ct to the gener ating set S := { a, b, a − 1 , b − 1 } is I. if z y x ≥ 0 , and I.1. if max( x 2 , y 2 ) ≤ | z | , then d ( e, g ) = 2  2 p | z |  − | x | − | y | ; I.2. if max( x 2 , y 2 ) ≥ | z | , and I.2.1. | xy | ≥ | z | , t hen d ( e, g ) = | x | + | y | ; I.2.2. | xy | ≤ | z | , t hen d ( e, g ) = 2 ⌈ min( | z /x | , | z /y | ) ⌉ +   | x | − | y |   ; II. if z y x ≤ 0 , and II.1 . if max( x 2 , y 2 ) ≤ | z | + | xy | , then d ( e, g ) = 2  2 p | z | + | xy |  − | x | − | y | ; II.2 . if max( x 2 , y 2 ) ≥ | z | + | xy | , then d ( e, g ) = 2 ⌈ min( | z /x | , | z /y | ) ⌉ + | x | + | y | . Given a word y , we call the pairs of consec utiv e letters of y the tr ansitions of y . The tra nsitions ab , ba − 1 , a − 1 b − 1 , a nd b − 1 a are said to b e p ositive. The transitions obtained by r e versing th e letters of these are c alled nega tive. Obser ve that the g roup element c orresp onding to a p ositive tra nsition is equa l to c times the element corresp onding to the negativ e transition obtained by reversing the order of the letters. So, if o ne tak es a w ord a nd transforms it b y reversing an equal n umber of positive a nd neg ative transitions , the resulting w ord is a geo desic if and only if the orig ina l was. Prop ositio n 5.2 . Every infinite ge o desic wor d in H 3 either has two elements of S as letters or c onsists of a fi nite pr efix fol lowe d by a single letter r ep e ate d. Pr o of. Let y be a n infinite word tha t has at least three elements of S as letter s , at least tw o of which appea r infinitely often. W e wis h to sho w that y is not a geodesic . Ob viously , this is the case if it is not fr eely reduced, so we a ssume the co n trary . Our ab ov e assumptions on y then imply that there are either infinitely man y p ositive tr ansitions in y o r ther e a r e infinitely ma ny negative transitions. Since y has at least three elements of S as letters, it must hav e as le tter s b oth some generato r and its inv erse. By rep eatedly swapping this gener ator with one of its neig hbours, we can bring it to a p ositio n adjacent to its inv erse. If y ha s bo th infinitely many positive and infinitely ma n y negative tra nsitions, then w e can compensate for these s waps b y reversing tra nsitions elsewhere so that the total num ber of p ositive transitions reversed e q uals the total num b er of nega tiv e ones r eversed. Since the resulting word is clearly not a geo desic, it follows that y is not either. 14 CORMA C W ALSH So we will as s ume without loss of generalit y that there are infi nitely many po sitive transitions a nd finitely many negative ones. Therefor e y can b e written in the form y = Ra i 0 b j 0 a − i 1 b − j 1 a i 2 · · · , where ( i n ) and ( j n ) a r e sequences of p ositive int egers, and R is some finite initial word. Consider wha t ha ppens if ( i n ) and ( j n ) even tually b ecome, respectively , con- stants i a nd j . Define the word w := a i b j a − i b − j . Obs erve that w = c ij . So, the word ww w · · · is no t geo de s ic since, by the formula of Theorem 5 .1, d ( e, w n ) grows like the square ro ot o f n , rather tha n linear ly . Therefo r e, in this ca se, y is not geo desic . If ( i n ) and ( j n ) are not b oth even tually cons tant , then we can find n ≥ 1 such that i n < i n +1 or j n < j n +1 . Assume that the latter holds; the former case can b e handled similarly . Also assume without loss of genera lity t hat n is even. So y can be wr itten y = R · · · b − j n − 1 ( a i n ) b j n a − i n +1 b − j n +1 · · · . By moving the br ack eted clump of a ’s forward, we get a word z = R · · · b − j n − 1 b j n a − i n +1 b − j n ( a i n ) b − j n +1 + j n · · · , which is clear ly no t a g eo desic. How ever, z is o btained from y by reversing the letters of i n j n po sitive transitions and then reversing the letters of the sa me num ber of negative transitio ns. It follows that y is no t geo desic either.  Theorem 5.3. L et ǫ a and ǫ b b e in {− 1 , +1 } , and let Ω b e the set of al l wor ds having letters in { a ǫ a , b ǫ b } with b oth letters o c cu rring infinitely often. Then, al l wor ds in Ω a r e ge o desic, and the c orr esp onding p aths al l c onver ge to t he same Busemann p oint, given by ξ ǫ a ǫ b ( c z b y a x ) := − ǫ a x − ǫ b y . Pr o of. Let w be a finite word having set o f letters { a ǫ a , b ǫ b } , and let w = c z b y a x be the as so ciated element o f the gr oup. If z 6 = 0, then sign z = sign y . sign x = ǫ b ǫ a . Therefore, z y x ≥ 0. Also, | z | ≤ | xy | . W e deduce fro m the word length formula of Theorem 5 .1 that d ( e, w ) = | x | + | y | , which is exactly the num b er of letters in w . So w is geo desic. It follows that every word in Ω is geo desic. Let w 1 and w 2 be words in Ω, and let γ 1 and γ 2 be the cor resp onding geo desic paths starting at the iden tit y . Choos e N ∈ N , and let u b e the prefix of w 1 of length N . Since w 2 has each of the letters a ǫ a and b ǫ b infinitely often, we can find a prefix v o f w 2 that has mo r e a ǫ a ’s and b ǫ b ’s than u . So there exists a word x with le tter s in { a ǫ a , b ǫ b } of leng th | v | − | u | s uch that ux = c m v for some m ∈ Z . Since w 2 has infinitely ma ny p ositive transitions a nd infinit ely man y negative tra nsitions, w e ca n write it w 2 = v y ω , where ω is an infinite word (in Ω) a nd y is a finite word sa tisfying y = c m z , for s o me z with le tters in { a ǫ a , b ǫ b } . By the first pa r t o f the prop osition, uxz ω is a geo desic word. The co rresp onding geo desic pa th agrees w ith γ 1 up to time N and agrees with γ 2 after time | uxz | . A similar arg ument shows that ther e is also a geo desic path agr eeing with γ 2 up AC TION OF A NILPOTENT GROUP ON ITS HOR OFUNCTION BOUNDAR Y 15 to time N and eventu ally coinciding with γ 1 . W e now apply Pr op osition 2.1 to deduce that γ 1 and γ 2 conv erge to the same limit. Consider th e seq ue nc e h n := ( a ǫ a b ǫ b ) n . By the previous part o f the pro po sition, this sequence converges to the commo n limit o f the geo desic s in Ω. W e hav e h n = c ǫ a ǫ b n ( n +1) / 2 b ǫ b n a ǫ a n . Let g := c z b y a x be a n arbitra ry p oint in H 3 . W e have d ( g , h n ) = d ( e, g − 1 h n ) = d ( e, c y x − z − ǫ b xn + ǫ a ǫ b n ( n +1) / 2 b ǫ b n − y a ǫ a n − x ) . Consider what happens whe n n is large. The exponents o f c , b , and a will have the signs ǫ a ǫ b , ǫ b , and ǫ a , resp ectively; therefore, their pr o duct will b e p ositive. The square of the exp onents of b oth a and b w ill b e approximately twice the absolute v alue of that of c , as will the abso lute v alue of the pro duct o f the exp onents o f a and b . Loo king at the formula of Theorem 5.1, we see that, f or lar ge n , the relev an t case is I.2.1 , a nd so d ( g , h n ) = | ǫ b n − y | + | ǫ a n − x | . In particular d ( e, h n ) = 2 n . So the limiting horofunction is ξ ǫ a ǫ b ( c z b y a x ) := − ǫ a x − ǫ b y .  Theorem 5.4 . Th e fol lowing fun ctions ar e Busemann p oints of H 3 : η ǫ m,n ( c k b j a i ) := − ǫi + | j − n | − | n | + 2 J  ǫ ( j − n ) , ( j − n ) i − ( k − m )  − 2 J ( − ǫn, m ) , ζ ǫ m,l ( c k b j a i ) := − ǫj + | i − l | − | l | + 2 J  − ǫ ( i − l ) , j l − ( k − m )  − 2 J ( ǫl , m ) , wher e ǫ ∈ {− 1 , + 1 } , m, n, l ∈ Z , and J ( u, v ) := ( 1 , if v 6 = 0 and u v ≥ 0 , 0 , otherwi se , for u, v ∈ Z . T o gether with ξ ++ , ξ − + , ξ ++ , and ξ −− , these ar e the only Busemann p oints. Pr o of. Cho ose ǫ ∈ {− 1 , +1 } and m, n ∈ Z , and define the path γ ǫ m,n ( t ) := c m b n a ǫt , for t ∈ Z . By the formula of Theo rem 5.1, d ( γ ǫ m,n (0) , γ ǫ m,n ( t )) = d ( e, a ǫt ) = t. Therefore γ ǫ m,n is a geo des ic starting a t c m b n . Let g := c k b j a i be a n element of H 3 . W e hav e d ( g , γ ǫ m,n ( t )) = d ( e, c m − k +( j − n ) i b n − j a ǫt − i ) . W rite x t := ǫt − i , y := n − j, a nd z := m − k + ( j − n ) i. So | x t | tends to infinity as t tends to infinity , whereas y and z remain co nstant. Therefore the only relev ant cases in the formula of Theo rem 5.1 when t is large are I.2 .1. a nd I I.2. If z = 0 , then b oth cas es give the sa me res ult: | x t | + | y | . If z 6 = 0 , then we have ca se I.2.1 if ǫy z > 0 , in whic h case d ( g , γ ǫ m,n ( t )) = | x t | + | y | , or case II.2 . if ǫy z ≤ 0, whic h gives d ( g , γ ǫ m,n ( t )) = | x t | + | y | + 2, for x t large enough. T o s um up: d ( g , γ ǫ m,n ( t )) = | x t | + | y | + 2 J ( − ǫy , z ) . 16 CORMA C W ALSH Subtracting fr o m this the v alue obtained when i = j = k = 0, and taking the limit as t tends to infinity , w e see that the path γ ǫ m,n conv erges to the ho rofunction η ǫ m,n defined a bove. This horo function is a Busemann po int since γ ǫ m,n is a geo des ic. Similar reaso ning shows that the pa th λ ǫ m,l ( t ) := c m + ǫtl b ǫt a l conv erges to ζ ǫ m,l , for all ǫ ∈ {− 1 , +1 } and m, l ∈ Z , and that the limit is a Busema nn p oint. By P rop osition 5.2, every infinite geo desic in H 3 either us es exactly tw o gen- erators or uses a ll but one o nly a finite num ber of times. In the former case , the geo desic con verges to one of ξ ++ , ξ − + , ξ + − , and ξ −− , b y Theorem 5.3. In the lat- ter, the geo desic must even tually coincide with either γ ǫ m,n or λ ǫ m,l for some v alue of ǫ a nd of m and n , or m and l . Ther e fore, in this case , the g eo desic conv erges to the corr esp o nding η ǫ m,n or ζ ǫ m,l .  The a ction of H 3 on each of the Busemann p oints can easily b e ca lculated: ( c z b y a x ) η ǫ m,n = η ǫ m + z + nx,n + y , ( c z b y a x ) ζ ǫ m,l = ζ ǫ m + z − y ( l + x ) ,l + x , ( c z b y a x ) ξ ǫ a ,ǫ b = ξ ǫ a ,ǫ b . So, in particular, e ach of the Busemann p oints ξ ++ , ξ + − , ξ − + , and ξ −− is fixed by the a ction of H 3 . W e ha ve here a n illus tr ation of Theo rem 1.1. The abelianisa tion of H 3 is isomorphic to Z 2 . One can take t he map φ to be φ ( c z b y a x ) := ( x, y ). The image o f S under this ma p is the set { (1 , 0) , (0 , 1) , ( − 1 , 0) , (0 , − 1) } , a nd s o P is the square with these po int s as cor ners. Of cour se, P ha s four facets, co rresp onding to the four fix e d Busema nn p oints. In [7], W ebs ter and Winchester conjectured that there is a b oundar y p oint or po in ts of the form lim i →∞ η + m i ,n i , where m i and n i grow without bound as i tends to infinity , with m i ≥ αn i even tua lly for an y α , and that this point or these p oints are fixed under the action of H 3 . Using the fo r mu la for η + m,n from T he o rem 5.4 one can calculate that ther e is exactly one p oint of this form, na mely ξ ++ , which is indeed a fix ed p oint. 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