Eigenvalue multiplicity and volume growth

Eigen v alue m ultiplicit y and v olume growth James R. Lee ∗ Univ ersity of W a shington Y ury Mak aryc hev Microsoft Res earch Abstract Let G be a finite group with symmetric gener ating set S , a nd let c = ma x R> 0 | B (2 R ) | | B ( R ) | be the doubling cons ta nt of the corre s po nding Cayley gra ph, where B ( R ) denotes an R -ball in the word-metric with respect to S . W e sho w that the mult iplicity of the k th eigenv alue of the Laplacian on the Ca y ley graph o f G is b ounded by a function of only c and k . Mo re sp ecifica lly , the m ultiplicit y is at most exp ( O (log c )(log c + log k )). Similarly , if X is a co mpact, n -dimensional Riemannia n manifold with non-neg ative Ricci curv ature, then the multiplicit y of the k th eigenv alue of the Laplace- Beltrami op erato r on X is at most exp ( O ( n )( n + log k ))). The fir st result (for k = 2) yields the following gro up-theoretic applicatio n. There exists a normal subgroup N of G , with [ G : N ] 6 α ( c ), and such that N admits a homomor phism onto Z M , where M > | G | δ ( c ) and α ( c ) 6 O ( h ) h 2 δ ( c ) > 1 O ( h log c ) , where h 6 exp((log c ) 2 ). This is a n effective, finitary analog of a theore m of Gromov which states that every infinite group of p olynomial gr owth has a subgroup of finite index which admits a homomorphism onto Z . This a ddr esses a question of T revisan, and is prov ed by scaling down Kleiner ’s pr o of o f Gromov’s theor em. In pa rticular, we replace the spac e of harmonic functions o f fixed p olynomial growth by the second eigenspace of the La placian o n the Cayley g raph of G . 1 In tro duction Let G b e a finitely generated group with finite, symmetric generating set S . The Ca yley graph Ca y ( G ; S ) is an undir ected | S | -regular graph with vertex set G and an edge { u, v } wh enev er u = v s for some s ∈ S . W e equip G with the natural w ord metric, whic h is also the shortest-path metric on Cay( G ; S ). Letting B ( R ) b e th e closed ball of r adius R about e ∈ G , one s a ys that G has p olynom ial gr owth if ther e exists a num b er m ∈ N suc h that lim R →∞ | B ( R ) | R m < ∞ . It is easy to see that this prop erty is in dep end en t of the c hoice of finite generating set S . ∗ Researc h supp orted by N SF CCF-06440 37. 1 In a classical p ap er [10], Gromo v p ro v ed that a group has p olynomia l growth if and only if it con tains a n ilp ote n t su bgroup of finite index. The sufficiency part w as pro ved earlier b y W olf [16]. It is natural to ask ab out similar phenomenon holds in finite gr oups . Of course, ev ery finite g roup has p olynomial gro wth tr ivially , so ev en formulating a similar question is not straigh tforwa rd. As Gromo v p oin ts out [10], by a compactness argument, one only needs | B ( R ) | 6 C R m to h old for R 6 R 0 , f or some R 0 = R 0 ( C, | S | , m ). Th us one can formulate a v ersion of Gromo v’s theorem for finite g roups. Ho w ever, there are n o effe ctiv e estimates kno wn for R 0 . F urthermore, Gromo v’s pro of relies on a limiting pro cedure whic h is again trivial for finite groups. Recen tly , Kleiner [13] ga ve a new pro of of Gromo v’s theorem that a voids the limiting pro cedur e, and in particular av oids the use of the Y amab e -Mon tgomery-Zipp in stru ctur e theory [14] to classify the limit ob j ects. The main step of Kleiner’s pro of lies in s h o w in g that the space of harmonic functions of fixed p olynomial gro wth is fi nite-dimensional on an infinite group G of p olynomial gro wth. Suc h a result follo ws from the w ork of Colding and Minicozzi [8] , but th eir pro of uses Gromo v’s theorem, whereas Kleiner is able to obtain the resu lt essen tially f rom scratc h, based on a new scale-dep endent Po incar ´ e inequalit y for b ounded-degree graphs. Again, the co nnection with finite groups is lac king: Ev ery harmonic f unction on a finite graph is constant . In the p resen t w ork, we s h o w that one can obtain some effe ctive p artial analogs of Gromo v’s theorem for finite groups by follo wing Kleiner’s general outline, but replacing the space of harmonic f unctions of fixed p olynomial gro wth with the second eigenspace of the d iscrete Laplacian on C a y ( G ; S ) . W e r ecall the follo win g tw o th eorems of Gromo v , w hic h captur e the essent ial m o ve from a geometric condition (p olynomial volume gro w th of balls) on an infinite group G , to a conclusion ab out its algebraic structure. Theorem 1.1 (Gromo v [10]) . If G is an infinite gr oup of p olynomial gr owth, the fol lo wing holds. 1. G admits a finite- dimensional line ar r epr esentation ρ : G → GL n ( C ) such that ρ ( G ) is infinite. 2. G c ontains a normal sub gr oup N , with [ G : N ] = O (1) , and such tha t N admits a homomor - phism onto Z . In fact, by the simp lifications of Tits (in App endix A.2 of [10]), Gromo v’s theorem follo ws fairly easily using an ind uction on (2). After seeing Kleiner’s pr o of, Luca T revisan asked whether there is a quant itativ e analog of part (1) o f Theorem 1.1 for finite groups. W e pro ve the follo wing. Theorem 1.2. L et G b e a finite gr oup. F or any symmetric g ener ating set S , define c = c G ; S = m ax R > 0 | B (2 R ) | | B ( R ) | , wher e B ( · ) is a close d b al l in Ca y( G ; S ) . Then the fol lo wing holds. 1. Ther e is a line ar r epr esentation ρ : G → GL ( R k ) , wher e k 6 exp  O (log c ) 2  , and | ρ ( G ) | > c − O (1) | G | 1 / log 2 ( c ) . 2. Ther e is a normal sub gr oup N 6 G , with [ G : N ] = O ( k ) k 2 , and N admits a homomorphism onto Z M , wher e M > c − O (1) | G | 1 / ( k log 2 c ) . 2 Observe that we hav e assumed a b ound on the r atios | B (2 R ) | / | B ( R ) | , which is stron ger than an assu mption o f the form | B ( R ) | 6 C R m for so me C, m > 0. (1) The latter t yp e of condition seems far more unwieldy in the setting of finite groups. By making suc h an a ssumption, w e completely b yp ass a “sc ale se lection” argument, and th e delicacy required b y Kleiner’s ap p roac h (wh ic h has to p erf orm man y steps of the pro of using only the geometry at a single scale). All of our argumen ts can b e carried out at a s ingle scale (see, e.g. t he R everse P oincar´ e Inequalit y for graphs in Section 3.1), but it is n ot clear whether th ere is an appropriate, effectiv e scale selection pro c edure in the finite case, and we lea v e the extension of Theorem 1.2 to a boun ded gro wth condition like (1) as an in teresting op en question. 1.1 Pro of outline and eigen v alue multipli city Our pr o of of Theorem 1.2 pr o ceeds along the follo wing lines. Giv en a n undirected d -regular graph H = ( V , E ), one d efines the discr ete L aplacian on H as the operator ∆ : L 2 ( V ) → L 2 ( V ) gi v en b y ∆( f )( x ) = f ( x ) − 1 d P y : { x,y }∈ E f ( y ). The eigen v alues of ∆ are non-negativ e and can b e ord ered λ 1 = 0 6 λ 2 6 · · · 6 λ n , where n = | V | . Th e se c ond eigensp ac e of ∆ is the subspace W 2 ⊆ L 2 ( V ) giv en b y W 2 = { f ∈ L 2 ( V ) : ∆ f = λ 2 f } . Finally , the w ell-kno wn (geometric) multiplicity of λ 2 is precisely dim ( W 2 ). In Section 3, we use the approac h of C olding and Minicozzi [8] and Kleiner [13] to argu e that dim ( W 2 ) = O (1), whenever c H = max x ∈ V , R > 0 | B ( x, 2 R ) | | B ( x,R ) | = O (1), and H satisfies a certain Poincar ´ e inequalit y . A t the h eart of the p ro of lies the intuition that fun ctions in W 2 are the “most h armonic- lik e” functions on H w hic h are orthogonal to the constan t functions. Carrying this out requires precise quan titativ e control on the eigenv a lues of H in terms of c H , whic h we obtain in Section 3.2. No w, consider H = Cay( G ; S ) for s ome fi n ite group G , an d the natural action of G on f ∈ L 2 ( G ) giv en by g f ( x ) = f ( g − 1 x ). It is easy to s ee that this ac tion comm utes with the action of the Laplacian, hence W 2 is an inv a rian t subsp ace. Since di m ( W 2 ) = O (1), w e will ha ve ac hiev ed Theorem 1.2(1) as long as t he image of the actio n is large. In S ection 4, we sho w that if the image of the action is small, then w e can pass to a sm all quotien t group, and that f pu shes do wn to an eigenfunction on the qu otien t. This allo ws u s to b ound λ 2 on the quotien t group in terms of λ 2 on G . But λ 2 on a small, co nnected graph cannot b e to o clo se to zero b y the d iscrete Ch eeger in equ alit y . In this wa y , we arrive at a co n tradiction if the image of the action is too small. Theorem 1 .2(2) is then a simple corol lary of Th eorem 1.2 (1), using a theorem o f Jordan on finite li near g roups. Higher eigen v alues a nd non-negativ ely curv ed ma nif olds. In fact, the tec hn iques of Section 3 giv e b oun ds on the multi plicit y of higher eigen v alues of the Laplacian as w ell, and the graph pro of extends r ather easily to b oun ding the eigen v alues of th e Laplace-B eltrami op e rator on Riemannian manifolds o f non-negativ e Ricci curv ature. Cheng [7] prov ed that the multiplici t y of the k th eigen v alue of a compact Riemannian surface of gen us g gro ws lik e O ( g + k + 1) 2 . Besson later sho wed [3 ] that the multi plicit y of the first non-zero eigen v alue is only O ( g + 1). W e refer to th e b ook of S c ho e n and Y au [15, Ch. 3] for further discussion of eigen v alue p roblems on manifolds. In Secti on 3, w e p ro ve a b ound on the m ultiplicit y of the k th smallest non-zero eigen v alue of the Laplace-Bel trami op erat or on compact Riemannian manifolds with non-negativ e Ricci curv atur e. In particular, the m ultiplicit y is b oun ded 3 b y a fu n ction dep end ing only on k and the d imension. Th e main add itional fact we require is an eigen v alue estimate of Ch eng [6] in this setting. 2 Preliminaries 2.1 Notation F or N ∈ N , w e write [ N ] for { 1 , 2 , . . . , N } . Giv en t wo expressions E and E ′ (p ossibly dep ending on a n umb er of parameters), w e w r ite E = O ( E ′ ) to mean that E 6 C E ′ for some constant C > 0 whic h is indep end en t of the parameters. Similarly , E = Ω( E ′ ) implies that E > C E ′ for some C > 0. W e also write E . E ′ as a synon ym for E = O ( E ′ ). Finally , we write E ≈ E ′ to denote the co njunction of E . E ′ and E & E ′ . In a m etric space ( X , d ), for a p o in t x ∈ X , w e use B ( x, R ) = { y ∈ X : d ( x, y ) 6 R } to denote the clo sed ball in X ab out x . 2.2 Laplacians , eigenv alues, and t he P oincar ´ e inequalit y Let ( X , dist , µ ) b e a metric-measure space. Throughout the pap e r, w e will b e in one of the follo wing t wo situat ions. ( G ) X is a finite, connected, und irected d -regular graph, di st is the shortest-path metric, and µ is the coun ting measure. In this case, we let E ( X ) denote the edge s et of X , and we write y ∼ x to denote { x, y } ∈ E ( X ). ( M ) X is a compact n -dimensional Riemannian manifold without b oundary , dist is the Riemannian distance, and µ is the Riemannian v olume. Since the pro ofs of S ection 3 pro ceed virtually id en tically in b oth cases, w e collect here some common notation. W e define k f k 2 = R f 2 dµ  1 / 2 for a fu nction f : X → R , and let L 2 ( X ) = L 2 ( X, µ ) b e the Hilb ert space of scalar functions for whic h k · k 2 is b ounded. In the graph setting, w e define the gradien t by [ ∇ f ]( x ) = 1 √ 2 d ( f ( x ) − f ( y 1 ) , . . . , f ( x ) − f ( y d )), w here y 1 , . . . , y d en u merate the n eigh b ors of x ∈ X . The actual order of enumeration is unimp ortan t as we will b e primarily concerned w ith the expression |∇ f ( x ) | 2 = 1 2 d X y : y ∼ x | f ( x ) − f ( y ) | 2 . W e define th e Sobolev space L 2 1 ( X ) =  f : Z f 2 dµ + Z |∇ f | 2 dµ < ∞  ⊆ L 2 ( X ) . No w w e pro ceed to define t he Laplac ian ∆ : L 2 1 ( X ) → L 2 1 ( X ). 1. In the graph setting, [∆ f ]( x ) = f ( x ) − 1 d X y : y ∼ x f ( y ). 2. In the Riemannian setting, ∆ is the Laplace-Beltrami op erator. It is w ell-known that in b oth our settings, ∆ is a self-adjoin t op erat or on L 2 1 ( X ) with discrete eigen v alues 0 = λ 1 < λ 2 6 λ 3 6 · · · . In the graph ca se, this sequence terminates with λ | X | . (Note 4 that we hav e used the graph-theoretic conv en tion for n um b erin g the eigenv a lues; in th e Riemannian setting, our λ 1 is usually written as λ 0 .) W e define th e k th eigensp ac e b y W k = { ϕ ∈ L 2 1 ( X ) : ∆ ϕ = λ k ϕ } in set ting ( G ), and W k = { ϕ ∈ L 2 1 ( X ) : ∆ ϕ + λ k ϕ = 0 } , in setting ( M ). The multiplicity of λ k is defined as m k = dim ( W k ). Observe the difference in s ign con ven tions, wh ich will not distur b us since we interac t with ∆ through the follo wing t wo facts. First, if λ is an eige n v alue of ∆ with co rresp o nding eige nfunction ϕ : X → R , then Z |∇ ϕ | 2 dµ = λ Z ϕ 2 dµ. (2) Secondly , b y the min -max prin ciple, if we ha v e fu nctions f 1 , f 2 , . . . , f k : X → R which ha v e m utually disjoin t supp o rts (and are thus linearly indep endent), then w e ha ve the b ound λ k 6 max i =1 ,...,k R |∇ f i | 2 dµ R ( f i − ¯ f i ) 2 dµ , (3) where ¯ f i = 1 µ ( X ) R f i dµ . I n the case k = 2, we actually need only a single test function f 1 : X → R in ( 3), since clearly f 1 − ¯ f 1 is orthogo nal to every constant function. The doubling condition. W e defin e c X = su p n µ ( B ( x, 2 R )) µ ( B ( x, R )) : x ∈ X, R > 0 o . Without loss of generalit y , and for the sak e of simplicit y , w e will assume th at c X > 2 throughout. Th e next theorem follo ws from standard v olume co mparison theorems (see, e.g. [12 ]). Theorem 2.1. In the setting ( M ) , if X has non-ne gative Ric ci curvatur e, then c X 6 2 n . The foll o w in g t wo f acts are straightfo rwa rd. F act 2.2. F or every ε, R > 0 , eve ry b al l of r adius R in X c an b e c over e d by c O (log( ε − 1 )) X b al l s of r adius εR . F act 2.3. If B = { B 1 , . . . , B M } is a disjoint c ol le ction of close d b al ls of r adius R , then the inter- se ction multiplicity of 3 B = { 3 B 1 , . . . , 3 B M } is at most c O (1) X . A Poinc ar´ e inequality . Finally , we d efine P X as the infim um o v er all num b e rs P for whic h the follo wing holds: F or ev ery R > 0, x ∈ X , and f : B ( x, 3 R ) → R , Z B ( x,R ) | f − ¯ f R | 2 dµ 6 P R 2 Z B ( x, 3 R ) |∇ f | 2 dµ, (4) where ¯ f R = 1 µ ( B ( x, R )) R B ( x,R ) f dµ . W e recall th e foll o w in g t wo kn o wn results about the relationship b et w een P X and c X . Theorem 2.4 (Kleiner and Saloff-Coste [13]) . In the setting ( G ) , if X is additio nal ly a Cayley gr aph, then P X . c 3 X . Theorem 2.5 (Buser [4]) . In the setting ( M ) , if X has non-ne gative Ric ci curvatur e, then P X . c X . 5 3 Eigen v a lue m ultiplicit y on doubling spaces In this section, w e pro v e the follo wing. Theorem 3.1. In b oth settings ( G ) and ( M ) , the multiplicity m k of the k th eigenvalue of the L aplacian on X sat isfies m 2 6 c O (log P X +log c X ) X , and m k 6 c O (log P X + c X log k ) X for k > 3 . If in the setting ( G ) , X is a Cayley gr aph, then for k > 2 , m k 6 exp ( O (log c X )(log c X + log k ) ) (5) If in the setting ( M ) , X additional ly has non-ne gative Ric ci curvatur e, then for k > 2 , m k 6 exp  O ( n 2 + n log k )  (6) W e will r equire th e foll o w ing eigen v alue b ound s. Theorem 3.2 (Chen g [6]) . In setting ( M ) , if X also has non-ne gative Ric ci cu rvatur e, then the k th eigenvalue of the L aplacian on X satifies λ k . k 2 n 2 diam ( X ) 2 . Cheng’s resu lt is prov ed via comparison to a m o del space of constan t sectional curv ature. In general, w e can p ro v e a w eake r b oun d u n der ju st a doub ling assu mption. The pr o o f is deferr ed to Section 3 .2. Theorem 3.3. In b oth settings ( G ) and ( M ) , the fol lo wing is true. The k th ei genvalue of the L aplacian on X sat isfies λ 2 6 c O (1) X diam ( X ) 2 , and λ k 6 k O ( c X ) diam ( X ) 2 for k > 3 . If, in addition, for every x, y ∈ X and R > 0 , we have µ ( B ( x, R )) = µ ( B ( y , R )) , then one obtains the estimate λ k . k 2 (log c X ) 2 diam ( X ) 2 , (7) for al l k > 2 . W e pro ceed to the proof of the theorem. Pr o of of The or em 3.1 . Let D = diam ( X ), and let B = { B 1 , B 2 , . . . , B M } b e a co ver of X of m in imal size b y balls of radius δD , for some δ > 0 to b e chosen later. By the doubling prop ert y (and F act 2.2), w e ha ve M 6 c O (log( δ − 1 )) X . Let W k b e the k th eigenspace of the Laplacian, and define the linear map Φ : W k → R M b y Φ j ( ϕ ) = 1 µ ( B j ) R B j ϕ dµ . Our goa l will b e t o show t hat f or δ > 0 small enough, Φ is injectiv e, and th us dim ( W k ) 6 M . 6 Lemma 3.4. If ϕ : X → R is a non-zer o eigenfunction of the L aplacian with eigenvalue λ 6 = 0 , and Φ( ϕ ) = 0 , then λ − 1 . c O (1) X P X ( δ D ) 2 . Pr o of. Using Φ j ( ϕ ) = 0 for e v ery j ∈ [ M ], and t he P oincar ´ e inequalit y (4), w e write Z ϕ 2 dµ 6 M X j =1 Z B j ϕ 2 dµ . P X ( δ D ) 2 M X j =1 Z 3 B j |∇ ϕ | 2 dµ. Also, M X j =1 Z 3 B j |∇ ϕ | 2 dµ 6 M (3 B ) Z |∇ ϕ | 2 dµ, where M (3 B ) = max x ∈ V # { j ∈ [ M ] : x ∈ 3 B j } 6 c O (1) X is the in tersection multiplic it y of 3 B (by F act 2.3). Combining th ese tw o inequalities and usin g (2) yields Z ϕ 2 dµ . c O (1) X P X ( δ D ) 2 Z |∇ ϕ | 2 dµ . c O (1) X P X ( δ D ) 2 λ Z ϕ 2 dµ whic h g iv es the desired conclusion. No w supp ose that ϕ ∈ W k and Φ( ϕ ) = 0. If ϕ 6 = 0, then by Lemma 3.4, w e ha v e λ k & c − O (1) X P − 1 X δ 2 diam ( X ) 2 . Cho osing δ > 0 sm all enough cont radicts Th eorem 3.2 or Theorem 3.3, d ep ending up on the as- sumption. It follo ws that dim ( W k ) 6 M 6 c O (log( δ − 1 )) X , yielding the desired b ounds. T o pr o ve (6), u s e Theorem 3.2, and observ e th at P X . c X b y Th eorem 2.5. T o obtain (5), observ e that P X . c 3 X , b y Theorem 2.4, and the condition of the eigen v alue estimate (7) is satisfied when X is a Cayle y graph (indeed, for an y v ertex-transitiv e graph). 3.1 Aside: A Reverse Poincar ´ e Inequalit y for graphs In the appr oac hes of Colding and Minicozzi [8] and Kleiner [13 ], one also needs a “rev erse P oincar ´ e inequalit y” to con trol harmonic fun ction on b alls, while in the preceding p ro of w e only n eed control of an eigenfunction on the en tire graph (for which w e could use (2)). W e observe the follo wing (p erhaps known) version for eigenfunctions on graph s. An analogous statement holds in setting (M) . Theorem 3.5. Supp o se we ar e in the gr aph setting (G) . L et ϕ : X → R b e an eigenfunction of the L aplac e op er ator with eigenvalue λ . Then, Z B ( R ) |∇ ϕ | 2 dµ 6  128 dR 2 + 2 λ  Z B (2 R ) ϕ 2 dµ. The proof is based on the follo wing lemma. 7 Lemma 3.6. L et ϕ : X → R b e an eigenf u nction of the L aplac e op er ator with e igenvalue λ . L et u : X → R b e a non-ne gative function that vanishes off B ( R − 1) , then Z B ( R ) u 2 |∇ ϕ | 2 dµ 6 128 d Z B ( R ) ϕ 2 |∇ u | 2 dµ + 2 λ k u k 2 ∞ Z B ( R ) ϕ 2 dµ. Pr o of. Denote S = 2 d R B ( R ) u |∇ ϕ | 2 dµ . Ass ume u v anishes o ff B ( R − 1). W e ha ve , S = Z B ( R ) X y ∼ x u ( x ) 2 | ϕ ( x ) − ϕ ( y ) | 2 dµ ( x ) = Z B ( R ) X y ∼ x u ( x ) 2 ( ϕ ( x ) 2 + ϕ ( y ) 2 − 2 ϕ ( x ) ϕ ( y )) dµ ( x ) = Z B ( R ) X y ∼ x u ( x ) 2 ϕ ( x ) 2 dµ ( x ) + Z B ( R ) X y ∼ x u ( x ) 2 ϕ ( y ) 2 dµ ( x ) − 2 Z B ( R ) X y ∼ x u ( x ) 2 ϕ ( x ) ϕ ( y ) dµ ( x ) = Z B ( R ) X y ∼ x u ( x ) 2 ϕ ( x ) 2 dµ ( x ) + Z B ( R ) X y ∼ x u ( y ) 2 ϕ ( x ) 2 dµ ( x ) − 2 Z B ( R ) X y ∼ x u ( x ) 2 ϕ ( x ) ϕ ( y ) dµ ( x ) = 2 Z B ( R ) X y ∼ x u ( x ) 2 ϕ ( x )( ϕ ( x ) − ϕ ( y )) dµ ( x ) + Z B ( R ) X y ∼ x ( u ( y ) 2 − u ( x ) 2 ) ϕ ( x ) 2 dµ ( x ) . Here we used that u ( x ) 2 ϕ ( x ) 2 = u ( y ) 2 ϕ ( x ) 2 = 0 when y / ∈ B ( R ), sin ce u v anishes off B ( R − 1). First, let us b ound the first term. Z B ( R ) X y ∼ x u ( x ) 2 ϕ ( x )( ϕ ( x ) − ϕ ( y )) dµ ( x ) = Z B ( R ) u ( x ) 2 ϕ ( x ) X y ∼ x ϕ ( x ) − ϕ ( y ) ! dµ ( x ) = d Z B ( R ) u 2 ϕ ∆ ϕ dµ = d Z B ( R ) u 2 λϕ 2 dµ 6 dλ k u k 2 ∞ Z B ( R ) ϕ 2 dµ. 8 No w w e b ound the second term. Z B ( R ) X y ∼ x ( u ( y ) 2 − u ( x ) 2 ) ϕ ( x ) 2 dµ ( x ) 6 2 Z B ( R ) X y ∼ x ( u ( y ) 2 − u ( x ) 2 )( ϕ ( x ) 2 − ϕ ( y ) 2 ) = 2 Z B ( R ) X y ∼ x | u ( x ) + u ( y ) | 2 | ϕ ( x ) − ϕ ( y ) | 2 dµ ( x ) ! 1 / 2 × Z B ( R ) X y ∼ x | u ( x ) − u ( y ) | 2 | ϕ ( x ) + ϕ ( y ) | 2 dµ ( x ) ! 1 / 2 6 8 Z B ( R ) X y ∼ x u ( x ) 2 | ϕ ( x ) − ϕ ( y ) | 2 dµ ( x ) ! 1 / 2 Z B ( R ) X y ∼ x | u ( x ) − u ( y ) | 2 | ϕ ( x ) | 2 dµ ( x ) ! 1 / 2 = 8 S 1 / 2 Z B ( R ) |∇ u | 2 ϕ 2 dµ ! 1 / 2 . Com b ining these b o unds we get, S 6 2 dλ k u k 2 ∞ Z B ( R ) ϕ 2 dµ + 8 S 1 / 2 Z B ( R ) |∇ u | 2 ϕ 2 dµ ! 1 / 2 . Therefore, e ither S 6 4 dλ k u k 2 ∞ Z B ( R ) ϕ 2 dµ and then we are d one, or S 6 16 S 1 / 2 Z B ( R ) |∇ u | 2 ϕ 2 dµ ! 1 / 2 . Then S 6 256 R B ( R ) |∇ u | 2 ϕ 2 dµ . Pr o of of The or em 3.5 . Th e theorem follo ws from Lemma 3.6, if we c ho ose u ( x ) = ( 1 , if x ∈ B ( R ) 1 − d ( x, B ( R )) /R, if x ∈ B (2 R ) \ B ( R ) . 3.2 Eigen v alue b ounds W e now pro ceed to pro v e the eigen v alue b ound s of Th eorem 3.3. Th e follo wing lemma is essen tially pro v ed in [11]; a similar statement with wo rse quan titativ e d ep endence can b e deduced f rom [2 ]. 9 Lemma 3.7. Ther e exi sts a c onstant A > 1 such that the f ol lowing holds. L et ( X , dist , µ ) b e any c omp act metric-me asur e sp ac e, wher e µ satisfies the doubling c ondition with c onstant c X . Then for any τ > 0 , ther e exists a finite p artition P of X into µ -me asur able subsets such that the fol lowing holds. If S ∈ P , then diam ( S ) 6 τ . F u rthermor e, if we u se P ( x ) denote the set P ( x ) ∈ P which c ontains x ∈ X , then µ  x ∈ X : di st ( x, X \ P ( x )) > τ A (1 + log( c X ))  > 1 2 . (8) Pr o of. W e w ill first defin e a r andom partitio n of X as follo ws . L et N = { x 1 , x 2 , . . . , x M } b e a τ / 4-net in X , and c ho ose a uniformly r andom bijection π : [ M ] → [ M ]. Also let α ∈ [ 1 4 , 1 2 ] b e c hosen uniform ly at random, and inductiv ely define S i = B ( x π ( i ) , ατ ) \ i − 1 [ j =1 S j . It is cle ar that P = S 1 ∪ S 2 ∪ · · · ∪ S M forms a p artition of X (note that some of the set s ma y b e empt y ), and diam ( S i ) 6 τ f or eac h i . Note that the d istr ibution of P is indep en d en t of th e measure µ . Claim 3.8. F or some A > 1 , and every x ∈ X , Pr P  dist ( x, X \ P ( x )) > τ A (1 + log ( c X ))  > 1 2 . (9) By a veraging, the claim implies that (8) holds for some p artition P of the requ ired form. F or the sak e of completeness, w e includ e here a simple pro of of C laim 3.8, whic h essentially follo ws from [5]. Pr o of of Claim 3.8. Fix a p oint x ∈ X and some v alue t 6 τ / 8. O bserve that, by F act 2. 2, we ha ve m = | N ∩ B ( x, τ ) | 6 c O (1) X . Order the p oin ts of N ∩ B ( x, τ ) in increasing distance from x : w 1 , w 2 , . . . , w m . Let I k = [ d ( x, w k ) − t , d ( x, w k ) + t ] and write E k for th e eve n t that ατ 6 d ( x, w k ) + t and w k is the min imal elemen t according to π for whic h ατ > d ( x, w k ) − t . It is straigh tforward to c heck that the ev ent { d ( x, X \ P ( x )) 6 t } is conta ined in the even t S m k =1 E k . T herefore, Pr [ d ( x, X \ P ( x )) 6 t ] 6 m X k =1 Pr[ E k ] = m X k =1 Pr[ ατ ∈ I k ] · Pr[ E k | ατ ∈ I k ] 6 m X k =1 2 t τ / 4 1 k 6 8 t τ (1 + log m ) , (10) where w e ha ve used the fact that Pr[ E k | ατ ∈ I k ] 6 Pr[min { π ( i ) : i = 1 , 2 , . . . , k } = π ( k )] = 1 /k . Th us choosing t ≈ τ 1+log( c X ) in ( 10) yields th e desired b ound (9). 10 The next simple le mma sho ws that on a “coarsely path-connected” space, a doubling mea sure cannot be concen trated on v ery small b alls. Lemma 3.9. L e t ( X, dist , µ ) satisfy ( G ) or ( M ) . Then for any x ∈ X and 10 6 R 6 D , w e have  1 − 1 2 c X  µ ( B ( x, R )) > µ ( B ( x, R / 10)) . Pr o of. Let δ = 1 / (2 c X ). Su pp ose there is an x ∈ X with µ ( B ( x, R/ 10)) > (1 − δ ) µ ( B ( x, R )), and 10 6 R 6 D . W e m a y assu me that µ ( B ( x, R )) = 1. In b oth settings ( G ) and ( M ), there exists a y ∈ X such that 3 R/ 5 > dist ( x, y ) > R/ 2. Let r = 3 R/ 8 so that B ( y , 2 r ) ⊇ B ( x, R/ 10) but B ( y, r ) ⊆ B ( x, R ) \ B ( x, R/ 10). In this case, µ ( B ( y , r )) 6 µ ( B ( x, R )) − µ ( B ( x, R/ 10)) 6 δ , and µ ( B ( y, 2 r )) = µ ( B ( y , 3 R/ 4)) > µ ( B ( x, R/ 10)) > 1 − δ > 1 − δ δ µ ( B ( y, r )) . Since (1 − δ ) /δ > c X , this violates the doubling assumption, yieldin g a con tradiction. Corollary 3.10. L et ( X , dist , µ ) satisfy ( G ) or ( M ) . Then for any x ∈ X and any ε > 0 , we have µ ( B ( x, ε di am ( X ))) 6 1 + µ ( X )  1 − 1 2 c X  O (log( ε − 1 )) . Under a symmetry assumption, there is an obvi ous impro vemen t. Lemma 3.11. L et ( X , dist , µ ) satisfy ( G ) or ( M ) . If , for every x, y ∈ X and R > 0 , we have µ ( B ( x, R )) = µ ( B ( y , R )) , then for eve ry ε > 0 , µ ( B ( x, ε di am ( X ))) . 1 + ε µ ( X ) . Pr o of. Fix x and y with di st ( x, y ) = diam ( X ), and connect x and y b y a geodesic γ . Let N ⊆ γ b e a maximal (3 ε di am ( X ))-separated set, so that | N | & 1 /ε . T hen the balls { B ( u, ε dia m ( X )) } u ∈ N are disjoin t, and eac h of e qual measure, implying the c laim. W e no w p ro v e Theorem 3. 3, yielding upp er b oun ds on the eigen v alues of ∆ . Pr o of of The or em 3.3 . Use Corollary 3.10 to choose τ > diam ( X ) e O ( c X log( k )) (11) so that for ev ery x ∈ X , µ ( B ( x, 2 τ )) 6 µ ( X ) 8 k . (12) Let P b e the p artition guarante ed b y Lemma 3.7 with p arameter τ . Since ev ery S ∈ P satisfies diam ( S ) 6 τ , (12) i mplies that µ ( S ) 6 µ ( X ) / (8 k ). Call a set S ⊆ X go o d if it satisfies µ  x ∈ S : dist ( x, X \ S ) > τ A (1 + log( c X ))  > 1 4 µ ( S ) . (13) 11 By a veraging, at least 1 / 4 of the measure is concen trated on go o d sets S ∈ P . In particular, since ev ery S ∈ P satisfies µ ( S ) 6 µ ( X ) / (8 k ), from the go o d sets S ∈ P , we can form (b y taking un ions of small sets) disjoin t set s S 1 , S 2 , . . . , S k suc h that eac h S i is go o d and satisfies µ ( X ) 8 k 6 µ ( S i ) 6 µ ( X ) 4 k . (14) No w defin e f i : X → R b y f i ( x ) = d ist ( x, X \ S i ). Clearly the f i ’s ha ve disjoin t supp o rt. F urther m ore, eac h f i is 1-Lipschitz, hence R |∇ f i | 2 dµ 6 µ ( X ) . Finally , since eac h set S i is go o d and sat isfies (14), we h a ve Z ( f i − ¯ f i ) 2 dµ & τ 2 (log c X ) 2 µ ( X ) k . Using (3), this implies that λ k 6 k O ( c X ) diam ( X ) 2 . Observe that w e can obtain a b et ter b ound λ 2 6 c O (1) X diam ( X ) 2 as follo ws. In this case, w e only need one test fu nction. Ch o ose τ = diam ( X ) / 20 ab o ve, and use Lemma 3.9 to form a goo d set S 1 whic h sat isfies µ ( X ) 2 c X 6 µ ( S 1 ) 6  1 − 1 2 c X  µ ( X ) , then define f 1 ( x ) = dist ( x, X \ S 1 ). Finally , to pro v e (7), n ote that under the measure symm etry assumption, w e can emplo y Lemma 3.11 to c h o ose τ > diam ( X ) O ( k ) in ( 11). The rest of the pro of pro ceeds exactly as b efore. 4 Applications to finite groups W e no w give some applications of Theorem 3.1 to finite g roups. Theorem 4.1. L et G b e a finite gr oup with symmetric gener ating set S . L et c G = m ax R> 0 | B (2 R ) | | B ( R ) | b e the doubling c onstant of the Cayley gr aph Cay( G ; S ) . Then ther e exists a finite-dimensional r epr esentation ρ W : G → GL ( W ) such that 1. dim W 6 exp ( O ( log c G ) 2 ) , 2. | ρ W ( G ) | & | G | 1 / log 2 c G /c O (1) G . Pr o of. Without loss o f ge neralit y , we assume that c G > 2 throughout. Rec all that d = | S | . Consider the action of G on L 2 ( G ) via [ ρ ( g ) f ]( x ) = f ( g − 1 x ). Note that th is action comm u tes with the Laplacian, [∆ ρ ( g ) f ]( x ) = [∆ f ]( g − 1 x ) = f ( g − 1 x ) − 1 d X s ∈ S f ( g − 1 xs ) = [∆ f ]( g − 1 x ) = [ ρ ( g )∆ f ]( x ) . 12 Therefore, ev ery eigenspace of the Laplaci an is in v ariant under th e ac tion of G . No w let W ≡ W 2 and λ 2 b e the second eigenspace and eigen v alue, resp ectiv ely , of the Laplacian on Cay( G ; S ). Let ρ W b e the restriction of ρ to W . First, b y Theorem 3.1(5), dim W 6 exp( O (log c G ) 2 ). No w w e need to pro ve the lo wer boun d on | ρ W ( G ) | . By Theorem 3.3, we ha ve λ 2 . c O (1) G diam (Ca y( G ; S )) 2 . (15) Consider H = ke r ρ W , the set of elemen ts whic h a ct trivially on W . H is a n ormal subgroup o f G a nd ρ W ( G ) ∼ = G/H . Let f b e an arbitrary non-zero fu nction in W 2 . Note that f is co nstan t on ev ery coset H g since the v alue of f ( hg ) = [ ρ ( h − 1 ) f ]( g ) = f ( g ) do es not dep end on h ∈ H . Define ˆ f : G/ H → R by ˆ f ( H g ) = f ( g ). Observe th at ˆ f is a non-constan t eigenfunction of the Laplacian on th e quotie n t graph Cay( G/H ; S ) with eig en v alue λ 2 , ∆ ˆ f ( H g ) = ˆ f ( H g ) − 1 d X s ∈ S ˆ f ( H g s ) = f ( g ) − 1 d X s ∈ S f ( g s ) = ∆ f ( g ) = λ 2 f ( g ) = λ 2 ˆ f ( H g ) . Let λ 2 ( G/H ) denote the second eigen v alue of the Laplacian on Ca y( G/H ; S ). Since λ 2 is a non-zero eigen v alue of th e Laplacian on C ay( G/H ; S ) , we hav e λ 2 ( G/H ) 6 λ 2 . Ho w ev er, by the discrete Cheege r inequalit y [1], λ 2 ( G/H ) > h (Ca y ( G/H ; S )) 2 2 d 2 (16) where h ( Ca y( G/H ; S )) is the Cheeger constan t of Cay( G/H ; S ) : h (Ca y ( G/H ; S )) ≡ max U ⊂ G/H ; | U | 6 | G/H | / 2 E ( U, ( G/H ) \ U ) | U | > 1 | G/H | / 2 , here E ( U, ( G/H ) \ U ) d enotes the set o f edges b et w een U and ( G/H ) \ U in Ca y ( G/H ; S ), and the b ound fol lo ws b ecause Ca y ( G/H ; S ) is a connecte d graph. W e conclude th at λ 2 > λ 2 ( G/H ) > (2 / | G/ H | ) 2 2 d 2 = 2 ( d | G/H | ) 2 . Com b ining this b oun d with (15), we get | ρ W ( G ) | = | G/H | > r 2 d 2 λ 2 & diam (Ca y( G ; S )) c O (1) G . The desired b o und n o w follo ws using the fact that diam ( Ca y ( G ; S )) > | G | 1 / log 2 c G . Corollary 4.2. Under the assumptions of The or em 4.1, ther e exists a normal sub gr oup N with [ G : N ] 6 α such that N has Z M as a homom orphic image, wher e M & | G | δ and δ = δ ( c G ) and α = α ( c G ) dep end only on the doubling c onstant of G . Pr o of. Let ρ W : G → GL ( W ) b e th e representati on guaran teed b y Theorem 4.1, and put k = dim W . Now, H = ρ W ( G ) is a fi nite su b group of GL ( W ), h ence by a theorem of Jordan (see [9, 36.13] ), H con tains a normal ab elian su bgroup A with [ H : A ] = O ( k ) k 2 . Since A is ab elian, its mem b ers can b e simultaneo usly diagonalized ov er C ; it follo ws that A is a pro d uct of at most k cyclic groups, hence Z M 6 A for some M > | A | 1 /k . 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