A new proof of Gromovs theorem on groups of polynomial growth

A NEW PROOF OF GR OMO V’S THEOREM ON GR OUPS OF P OL YNOMIAL GR O WTH BRU CE KLEINER Abstract. W e give a proo f o f Gromov’s theorem that an y finitely generated group of polynomial growth ha s a finite index nilp o ten t subgroup. The pro of do es not rely on the Montgomery-Zippin- Y a mab e structure theory of lo cally compa c t gro ups. Contents 1. In tro duction 1 2. A P oincare inequalit y for finitely generated gr oups 3 3. The pro of of Theorem 1.4 5 4. Pro of of Corolla ry 1.5 using Theorem 1.4 10 5. Pro of of Corolla ry 1.6 using Corollary 1.5 11 App endix A. Prop erty (T) and equiv arian t harmonic maps 12 References 15 1. Introduction 1.1. Statemen t of results. Definition 1.1. Let G be a finitely generated group, and let B G ( r ) ⊂ G d enote the ball cen t ered at e ∈ G with resp ect to some fixed w or d norm on G . The g roup G has p olynomial growth if for some d ∈ (0 , ∞ ) (1.2) lim s up r →∞ | B G ( r ) | r d < ∞ , Date : May 30, 2018. Suppo rted b y NSF Grant DMS 0 70151 5. 1 2 BRUCE KLEIN ER and has wea kly p olynomial growth if for some d ∈ (0 , ∞ ) (1.3) lim inf r →∞ | B G ( r ) | r d < ∞ , W e giv e a pro of of the follow ing sp ecial case o f a theorem of Colding- Minicozzi, without using G romo v’s theorem on groups of p o lynomial gro wth: Theorem 1.4 ([CM97]) . L et Γ b e a Cayley gr aph of a gr o up G of we akly p olynomial gr owth, and d ∈ [0 , ∞ ) . Then the sp ac e of harmonic functions on Γ with p olynomial gr o w th at most d is finite dimens i o nal. Note that although [CM97 ] stated the result for groups o f p olynomial gro wth, their pr o of also w orks for groups of w eakly p olynomial g r o wth, in view of [vdDW84]. W e then use this to deriv e the following corollaries: Corollary 1.5. If G is an i n finite gr oup of we akly p olynomial gr o wth, then G admits a finite d imensional line ar r epr esentation G → GL ( n, R ) with infinite im age. Corollary 1.6 ( [Gro8 1, vdD W84]) . If G is a gr oup w ith we akly p oly- nomial gr owth, then G is virt ual ly nilp otent. W e emphasize that our pro of of Corolla ry 1.6 yields a new proof o f Gromov ’s theorem on groups o f p olynomial gro wth, whic h do es not in- v olve the Montgomery -Zippin-Y amabe structure theory of lo cally com- pact gro ups [MZ74]; ho w eve r, it still relies on Tits’ alternative for linear groups [T it72] (or the easier the orem of Shalom that amenable linear groups are virtually solv able [Sha98 ]) . R emark 1.7 . There ar e sev eral imp o r t an t applications of the Wilkie- V an Den Dries refinemen t [vdDW84] of Gromov’s theorem [Gro81] that do not fo llo w fro m the orig inal statemen t; for instance [Pap05], or the theorem of V a rop oulos that a group satisfies a d -dimensional Euclidean isop erimetric inequalit y unless it is virtually nilp oten t of gro wth ex p o- nen t < d . 1.2. Sk et ch of the pro ofs. The pro of of Theorem 1.4 is based on a new Poincare inequalit y whic h holds for an y Cayle y graph Γ o f an y finitely generated group G : (1.8) Z B ( R ) | f − f R | 2 ≤ 8 | S | 2 R 2 | B (2 R ) | | B ( R ) | Z B (3 R ) |∇ f | 2 , GROUPS OF POL YNOMIAL GRO WTH 3 Here f is a pie cewise s mo oth func tion on B (3 R ), f R is the a v era g e of u ov er the ball B ( R ), and S is the g enerating set for G . The remainder of the pro of has the same rough outline as [C M97], though the de tails are differen t. Note that [CM97] a ssumes a uniform doubling condition as w ell as a uniform P oincare inequalit y . In our con text, w e ma y not app eal to suc h uniform b o unds as their pro of dep ends on Gromo v’s theorem. Instead, the idea is to use (1.8) to sho w that one has uniform b ounds at certain scales, and that this is sufficien t to de duce that the space of harmonic functions in question is finite dimensional. The pro of of Corollary 1.5 inv ok es a Theorem o f [Mok95 , KS97] to pro duce a fixed p o in t free isometric G -action G y H , where H is a Hilb ert space, and a G -equiv arian t harmonic map f : Γ → H from the Ca yley graph of G to H . Theorem 1.4 then implies that f take s v alues in a fi nite dimens ional subspace of H , and this implies Corollary 1.5. See Section 4. Corollary 1.6 f o llo ws from Corollary 1 .5 b y induction on the degree of grow th, as in the origina l pro of of G romo v; see Section 5. 1.3. Ac kno w ledgemen ts. I would lik e to thank Ala in V a lette for an inspiring lecture at MSRI in August 2007, a nd the discussion after- w ard. This gav e me the initia l imp etus to find a new pro of of G ro- mo v’s theorem. I w ould esp ecially lik e to thank Lauren t Saloff-Coste for telling me ab out the P o incare inequalit y in Theorem 2.2, whic h has replaced a more complicated o ne used in an earlier draft of this pap er, and Bill Minicozzi f or simplifying Section 3. Finally I w a n t to thank T oby Colding for sev eral con v ersations regarding [CM97], and Emmaneul Breuillard, David Fis her, Misha Kap o vic h, Bill Minicozzi, Lior Silb erman a nd Alain V alette for commen ts and corrections. 2. A Poincare inequality f or finitel y ge nera ted groups Let G b e a gro up, with a finite generating set S ⊂ G . W e denote t he asso ciated w ord no r m of g ∈ G b y | g | . F o r R ∈ [0 , ∞ ) ∩ Z , let V ( R ) = | B G ( R ) | = | B G ( e, R ) | . W e will denote the R -ball in the associated Ca yley graph by B ( R ) = B ( e, R ). R emark 2.1 . W e are viewing t he Cay ley graph as (the geometric real- ization of a) 1-dimensional simplicial complex, not as a discre te space. Th us B G ( R ) is a finite set, whereas B ( R ) is t ypically 1-dimensional. 4 BRUCE KLEIN ER Theorem 2.2. F or every R ∈ [0 , ∞ ) ∩ Z and every sm o oth function f : B (3 R ) → R , (2.3) Z B ( R ) | f − f R | 2 ≤ 8 | S | 2 R 2 V (2 R ) V ( R ) Z B (3 R ) |∇ f | 2 , wher e f R is the aver age of f over B ( R ) . Pr o of. Fix R ∈ [0 , ∞ ) ∩ Z . Let δ f : B G (3 R − 1) → R b e given b y δ f ( x ) = Z B ( x, 1) |∇ f | 2 . F or ev ery y ∈ G , w e c ho ose a shortest vertex path γ y : { 0 , . . . , | y | } → G from e ∈ G to y . If y ∈ B G (2 R − 2), then (2.4) X x ∈ B ( R − 1) | y | X i =0 ( δ f )( x γ y ( i )) ≤ 2 R X z ∈ B (3 R − 1) ( δ f )( z ) , since the map B ( R − 1) × { 0 , . . . , | y |} → B (3 R − 1) given b y ( x, i ) 7→ x γ y ( i ) is at most 2 R -to-1. F or ev ery ordered pair ( e 1 , e 2 ) of edges con tained in B ( R ), let x i ∈ e i ∩ G b e elemen ts such that d ( x 1 , x 2 ) ≤ 2 R − 2, a nd let y = x − 1 1 x 2 . By the Cauch y-Sc hw arz in equalit y , (2.5) Z ( p 1 ,p 2 ) ∈ e 1 × e 2 | f ( p 1 ) − f ( p 2 ) | 2 dp 1 dp 2 ≤ 2 R | y | X i =0 ( δ f )( x 1 γ y ( i )) . No w Z B ( R ) | f − f R | 2 ≤ 1 V ( R ) Z B ( R ) × B ( R ) | f ( p 1 ) − f ( p 2 ) | 2 dp 1 dp 2 = 1 V ( R ) X ( e 1 ,e 2 ) ⊂ B ( R ) × B ( R ) Z ( p 1 ,p 2 ) ∈ e 1 × e 2 | f ( p 1 ) − f ( p 2 ) | 2 dp 1 dp 2 ≤ 1 V ( R ) X ( e 1 ,e 2 ) ⊂ B ( R ) × B ( R ) 2 R | y | X i =0 ( δ f )( x 1 γ y ( i )) , where x 1 and y are as defined ab ov e. T he map ( e 1 , e 2 ) 7→ ( x 1 , y ) is at most | S | 2 -to-one, so Z B ( R ) | f − f R | 2 ≤ 2 R | S | 2 1 V ( R ) X x 1 ∈ B ( R − 1) X y ∈ B (2 R − 2) | y | X i =0 ( δ f )( x 1 γ y ( i )) GROUPS OF POL YNOMIAL GRO WTH 5 ≤ 4 R 2 | S | 2 1 V ( R ) X y ∈ B (2 R − 2) X z ∈ B (3 R − 1) ( δ f )( z ) (b y (2.4 ) ) = 4 R 2 | S | 2 V (2 R ) V ( R ) X z ∈ B (3 R − 1) ( δ f )( z ) ≤ 8 R 2 | S | 2 V (2 R ) V ( R ) Z B (3 R ) |∇ f | 2 .  R emark 2.6 . Altho ugh the theorem ab ov e is no t in the literat ur e, the pro of is virtually con tained in [CSC93 , pp.308 - 310]. When hearing of m y more complicated P oincare inequality , Laur ent Saloff-Coste’s immediate resp onse w as to state and prov e Theorem 2.2. 3. The proo f of Theorem 1.4 In this section G will b e a finitely generated group with a fixed finite generating set S , and t he as so ciated Cayley gr a ph and w ord norm will b e denoted Γ and k · k , r espectiv ely . F o r R ∈ Z + w e let B ( R ) := B ( e, R ) ⊂ Γ and V ( R ) := | B G ( R ) | = | B ( R ) ∩ G | . Let V b e a 2 k -dimensional v ector space o f harmonic functions on Γ. W e equip V with the family of quadratic fo rms { Q R } R ∈ [0 , ∞ ) , where Q R ( u, u ) := Z B ( R ) u 2 . The remainder of this section is dev ot ed to pro ving the f ollo wing statemen t, whic h clearly implies Theorem 1.4: Theorem 3.1. F or eve ry d ∈ (0 , ∞ ) ther e is a C = C ( d ) ∈ (0 , ∞ ) such that if (3.2) lim inf R →∞ V ( R ) (det Q R ) 1 dim V R d < ∞ , then dim V < C . The o v erall structure of the pro of is similar to that of Colding- Minicozzi [CM97]. 3.1. Finding go o d scales. W e b egin b y using the p olynomial growth assumption to select a pair of compara ble scales R 1 < R 2 at which b o th the growth function V and the determinan t (det Q R ) 1 dim V ha v e doubling b eha vior. Lat er w e will use this to find many functions in V whic h ha v e doubling b eha vior a t scale R 2 . Similar scale selection argumen ts app ear in b oth [G ro81] and [CM97]; t he one here is a h ybrid of the tw o. 6 BRUCE KLEIN ER Observ e that the family of quadratic forms { Q R } R ∈ [0 , ∞ ) is nonde- creasing in R , in the sense that Q R ′ − Q R is p ositiv e semi-definite when R ′ ≥ R . Also, note that Q R is p ositiv e definite fo r sufficien tly lar g e R , since Q R ( u, u ) = 0 for a ll R only if u ≡ 0. Cho ose i 0 ∈ N suc h that Q R > 0 whenev er R ≥ 16 i 0 . W e define f : Z + → R and h : Z ∩ [ i 0 , ∞ ) → R by f ( R ) = V ( R ) (det Q R ) 1 dim V , and h ( i ) = log f (16 i ) . Note tha t since Q R is a nondecreasing function of R , b oth f and h are nondecreasing functions, a nd (3.2) tra nslates to (3.3) lim inf i →∞ ( h ( i ) − di log 16) < ∞ . Put a = 4 d log 16, and pic k w ∈ N . Lemma 3.4. Ther e ar e inte gers i 1 , i 2 ∈ [ i 0 , ∞ ) such t hat (3.5) i 2 − i 1 ∈ ( w , 3 w ) , (3.6) h ( i 2 + 1) − h ( i 1 ) < w a, and (3.7) h ( i 1 + 1) − h ( i 1 ) < a, h ( i 2 + 1) − h ( i 2 ) < a. Pr o of. There is a nonnegative in t eger j 0 suc h that (3.8) h ( i 0 + 3 w ( j 0 + 1)) − h ( i 0 + 3 w j 0 ) < w a. Otherwise, for all l ∈ N w e would get h ( i 0 + 3 w l ) = h ( i 0 ) + l − 1 X j = 0 ( h ( i 0 + 3 w ( j + 1)) − h ( i 0 + 3 w j )) ≥ h ( i 0 ) + w al = h ( i 0 ) +  4 3 d log 16  (3 w l ) , whic h contradicts (3.3) for larg e l . Let m := i 0 + 3 w j 0 . Then there are in tegers i 1 ∈ [ m, m + w ) and i 2 ∈ [ m +2 w , m + 3 w ) suc h that (3.7) ho lds, for otherwise w e would ha v e either h ( m + w ) − h ( m ) ≥ w a or h ( m + 3 w ) − h ( m + 2 w ) ≥ w a , con tradicting (3.8 ) . These i 1 and i 2 satisfy the conditions of the lemma, b ecause h ( i 2 + 1) − h ( i 1 ) ≤ h ( m + 3 w ) − h ( m ) < w a.  GROUPS OF POL YNOMIAL GRO WTH 7 3.2. A con trolled co ver. Let R 1 = 2 · 16 i 1 and R 2 = 16 i 2 . Cho o se a maximal R 1 -separated subset { x j } j ∈ J of B ( R 2 ) ∩ G , and let B j := B ( x j , R 1 ). Then the collection B := { B j } j ∈ J co vers B ( R 2 ), and 1 2 B := { 1 2 B j } j ∈ J is a disjoint collection. Lemma 3.9. (1) The c overs B and 3 B := { 3 B j } j ∈ J have interse c- tion multiplicity < e a . (2) B has c ar dinality | J | < e w a . (3) Ther e is a C ∈ ( 0 , ∞ ) dep ending only on | S | such that for eve ry j ∈ J and every smo oth func tion v : 3 B i → R , (3.10) Z B i | v − v B i | 2 ≤ C e a R 2 1 Z 3 B i |∇ v | 2 . Pr o of. (1) If z ∈ 3 B j 1 ∩ . . . ∩ 3 B j l , then x j m ∈ B ( x j 1 , 6 R 1 ) for ev ery m ∈ { 1 , . . . , l } , so { B ( x j m , R 1 2 ) } l m =1 are disjoint balls lying in B ( x j 1 , 8 R 1 ), and hence log l ≤ lo g V (3 R 1 ) V ( R 1 2 ) = log V (3 R 1 ) − lo g V  R 1 2  ≤ h ( i 1 + 1) − h ( i 1 ) < a. This sho ws that the multiplicit y of 3 B is at most e a . This implies (1), since the multiplicit y of B is not greater than t ha t of 3 B . (2) The balls { B ( x j , R 1 2 ) } j ∈ J are disjoin t, and are con tained in B ( R 2 + R 1 2 ) ⊂ B (2 R 2 ), so | J | ≤ V (2 R 2 ) V ( R 1 2 ) ≤ V (16 i 2 +1 ) V (16 i 1 ) < e w a , b y (3.6). (3) By Theorem 2.2 and t he translation inv ariance of the inequ alit y , Z B i | v − v B i | 2 ≤ 8 | S | 2 R 2 1 V (2 R 1 ) V ( R 1 ) Z 3 B i |∇ v | 2 ≤ 8 | S | 2 R 2 1 e a Z 3 B i |∇ v | 2 .  8 BRUCE KLEIN ER 3.3. Estimating functions relative to the co v er B . W e no w esti- mate t he siz e o f a harmonic function in terms of it s a v erag es o ve r the B j ’s, and its size on a larger ball. W e define a linear map Φ : V → R J b y Φ j ( v ) := 1 | B j | Z B j v . Lemma 3.11 (cf. [CM97, Pro p. 2.5 ]) . Ther e is a c onstant C ∈ (0 , ∞ ) dep ending only on the size o f the gener ating set S , w i th the fol lowing pr op erty. (1) If u is a smo oth functions on B (16 R 2 ) , then (3.12) Q R 2 ( u, u ) ≤ C V ( R 1 ) | Φ( u ) | 2 + C e 2 a R 2 1 Z B (2 R 2 ) |∇ u | 2 . (2) If u is ha rm onic on B (16 R 2 ) , then (3.13) Q R 2 ( u, u ) ≤ C V ( R 1 ) | Φ( u ) | 2 + C e 2 a  R 1 R 2  2 Q 16 R 2 ( u, u ) . Pr o of. W e will use C to denote a constant whic h dep ends only on | S | ; ho w ev er, its v alue ma y v ary fro m equation to equation. W e hav e Q R 2 ( u, u ) = Z B ( R 2 ) u 2 ≤ X j ∈ J Z B j u 2 (3.14) ≤ 2 X j ∈ J Z B j  | Φ j ( u ) | 2 + | u − Φ j ( u ) | 2  . W e estimate eac h of the terms in (3.14) in turn. F or the first term w e get: (3.15) X j ∈ J Z B j | Φ j ( u ) | 2 = X j ∈ J | B j | | Φ j ( u ) | 2 ≤ C V ( R 1 ) | Φ( u ) | 2 . F or the second t erm w e hav e: X j ∈ J Z B j | u − Φ j ( u ) | 2 ≤ C e a R 2 1 X j ∈ J Z 3 B j |∇ u | 2 (b y (3) of L emma 3.9 ) GROUPS OF POL YNOMIAL GRO WTH 9 ≤ C e a R 2 1  e a Z B (2 R 2 ) |∇ u | 2  (b y (1) of L emma 3.9) = C e 2 a R 2 1 Z B (2 R 2 ) |∇ u | 2 . Com bining this with (3.15) yields (1). Inequalit y (3.13 ) follow s from (3.12) b y applying the rev erse P oincare inequalit y , whic h holds for a ny harmonic f unction v defined on B (16 R 2 ): R 2 2 Z B (2 R 2 ) |∇ v | 2 ≤ C Q 16 R 2 ( v , v ) . (F o r the pro of, see [SY95, Lemma 6.3], and note that for harmonic functions their condition u ≥ 0 may be dropp ed.)  3.4. Selecting functions from V with con trolled gro wth. Our next step is to select functions in V whic h ha ve d oubling b eha vior at scale R 2 . Lemma 3.16 (cf. [CM97, Prop. 4.16]) . Th e r e is a subsp ac e U ⊂ V of dimension at le as t k = dim V 2 such that fo r every u ∈ U (3.17) Q 16 R 2 ( u, u ) ≤ e 2 a Q R 2 ( u, u ) . Pr o of. Since R 2 = 16 i 2 > 16 i 0 , the quadratic fo r m Q R 2 is p ositiv e definite. Therefore there is a Q R 2 -orthonormal basis β = { v 1 , . . . , v 2 k } for V whic h is orthog onal with resp ect to Q 16 R 2 . Supp ose there are at least l distinct elemen ts v ∈ β s uc h that Q 16 R 2 ( v , v ) ≥ e 2 a . Then since β is Q R 2 -orthonormal and Q 16 R 2 -orthogonal, log  det Q 16 R 2 det Q R 2  1 2 k = log 2 k Y j = 1 Q 16 R 2 ( v j , v j ) Q R 2 ( v j , v j ) ! 1 2 k = log 2 k Y j = 1 Q 16 R i ( v j , v j ) ! 1 2 k ≥ log  e 2 al  1 2 k = l k a. On the other hand, a > h ( i 2 + 1) − h ( i 2 ) ≥ lo g (det Q 16 R 2 ) 1 2 k − log (det Q R 2 ) 1 2 k . So w e hav e a con tradiction if l ≥ k . Therefore w e may c ho ose a k elemen t subset { u 1 , . . . , u k } ⊂ { v 1 , . . . , v 2 k } suc h that Q 16 R 2 ( u j , u j ) < e 2 a for ev ery j ∈ { 1 , . . . , k } . Then every ele- men t of U := span { u 1 , . . . , u k } satisfies ( 3.17).  10 BRUCE KLEIN ER 3.5. Bounding the dimension of V . W e now assume that w is the smallest inte ger suc h that (3.18)  R 1 R 2  2 = 2 · 16 i 1 − i 2 < 2 · 16 − w < 1 2 C e 4 a , where C is the c onstan t in (3.13) . Therefore 2 · 16 − ( w − 1) ≥ 1 2 C e 4 a , and this implies (3.19) e w a ≤ 64 C e 64 d 2 log 16 . If u ∈ U lies in the k ernel of Φ, then Q R 2 ( u, u ) ≤ C e 2 a  R 1 R 2  2 Q 16 R 2 ( u, u ) (b y (3.13) ) ≤ C e 2 a  R 1 R 2  2  e 2 a Q R 2 ( u, u )  (b y Lemma 3.16) ≤ 1 2 Q R 2 ( u, u ) (b y (3 .1 8) ) . Therefore u = 0, and w e conclude that Φ | U is injectiv e. Hence b y Lemma 3.9 and (3.19), dim V = 2 dim U ≤ 2 | J | ≤ 2 e w a ≤ 1 28 C e 64 d 2 log 16 .  4. Proof o f Corollar y 1.5 using Theorem 1.4 Let G b e as in the statemen t o f the Coro llary , a nd let Γ denote some Ca yley graph of G with respect to a symmetric finite generating set S . Note that G is amenable, for if R k → ∞ and V ( R k ) < AR d k for all k , then for ev ery k there m ust b e an r k ∈ [ R k 2 , R k ] suc h that the ball B G ( r k ) satisfies | ∂ B G ( r k ) | = | S G ( r k ) | < 3 A R d − 1 k ; this means that the sequence of balls { B G ( r k ) } is a F olner sequence for G . Hence G do es not hav e P rop erty (T). Th erefore by a re sult o f Mok [Mok95] and Korev aar-Sc ho en [KS97, Theorem 4.1.2], there is an iso- metric action G y H of G on a Hilb ert space H whic h has no fixed GROUPS OF POL YNOMIAL GRO WTH 11 p oin ts, and a nonconstan t G - equiv arian t harmonic map f : Γ → H . In the case of Ca yley graphs, the Mok/Korev aar-Sch o en result is quite elemen tary , so w e g iv e a short pro of in App endix A. Since f is G -equiv a r ian t, it is Lipsc hitz. Eac h b ounded linear functional φ ∈ H ∗ giv es rise to a Lipsc hitz harmonic function φ ◦ f , and hence w e ha v e a linear map Φ : H ∗ → V , where V is the space of Lipsc hitz harmonic functions o n Γ. Since the target is finite dimensional b y Th eorem 1.4, the k ernel of Φ has finite co dimension, and its a nnihilat o r k er (Φ) ⊥ ⊂ H is a finite dimensional subspace containing the image of f . It fo llo ws that the affine h ull A of the imag e of f is finite dimensional and G -inv ariant. Therefore w e hav e an induced isometric G -action G y A . This action cannot factor through a finite group, b ecause it w ould then hav e fixed p o in ts, con tr a dicting the fact that the o riginal r epresen tation is fixed p oin t free. The ass o ciated homomorphism G → Isom( A ) yields the des ired finite dimensional represen tation of G .  5. Proof o f Cor ollar y 1.6 using Corolla r y 1.5 W e prov e Gromov ’s theorem using Corollary 1.5. The pro of is a recapitulation of Gro mo v’s argumen t, whic h repro duce here for the con venie nce of the reader. The pro o f is b y induction on the degree of g r o wth. Definition 5.1. Let G b e a finitely generated group. The degree ( of gro wth) of G is the minim um deg( G ) of t he nonnegative in tegers d suc h that lim inf r →∞ V ( r ) r d < ∞ . A group whose degree of gro wth is 0 is finite, and hence Corollary 1.6 holds for suc h a gro up. Assume inductiv ely that for some d ∈ N that eve ry group of degree at most d − 1 is virtually nilp oten t, and supp ose deg( G ) = d . Then G is infinite, and b y Corollary 1.5 there is a finite dimensional linear represen tation G → GL ( n ) with infinite image H ⊂ GL ( n ). Since H has p olynomial growth, by [Tit72 ] ( see [Sha9 8 ] for an easier proof ) it is virtually solv able, and b y [W o l6 8, Mil68] it m ust b e virtually nilp otent. After passing to finite index subgroups, w e may assume H is n ilp o- ten t, and that its ab elianization is torsion-free. It follows that there is 12 BRUCE KLEIN ER a short exact sequence 1 − → K → G α → Z − → 1 . By [vdD W84, Lemma (2.1)], the normal subgroup K is finitely gener- ated, and deg ( K ) ≤ deg ( G ) − 1. By the induction hy p othesis, K is virtually nilpot ent. Let K ′ b e a finite index nilpotent su bgroup of K whic h is normal in G , and let L ⊂ G b e an infinite cyclic subgroup which is mapped isomorphically b y α on to Z . Then K ′ L ⊂ G is a finite index solv able s ubgroup of G . As it has p olynomial gro wth, by [W ol68, Mil68] it is virtually nilp otent.  Appendix A. Proper ty (T) and equiv ariant harmonic maps In this exp ository section, w e w ill giv e a simple pro of of the sp ecial case of the Ko rev aar- Sc ho en/Mok existence result needed in the pro of of Corollary 1 .6. Supp ose G is a finitely generated gro up, S = S − 1 ⊂ G is a symmetric finite generating set, and Γ is the asso ciated Cay ley gra ph. Giv en an action G y X on a metric space X , w e define the energy function E : X → R by E ( x ) = X s ∈ S d 2 ( sx, x ) . W e recall that a G has P rop erty (T) iff eve ry isometric action of G on a Hilb ert space has a fixed p o int. The following theorem is a very w eak v ersion of some res ults in [FM05], see also [G ro03, pp.115- 1 16]: Theorem A.1. The fol lowing ar e e quiva lent: (1) G has Pr op erty (T). (2) Ther e is a c onstant D ∈ (0 , ∞ ) such that if G y H is an isometric action on a Hilb ert sp ac e and x ∈ H , then G fixes a p oint in B ( x, D p E ( x )) . (3) Ther e ar e c ons tants D ∈ ( 0 , ∞ ) , λ ∈ (0 , 1) such that if G y H is an isometric action on a Hilb ert sp ac e and x ∈ H , then ther e is a p oin t x ′ ∈ B ( x, D p E ( x )) such that E ( x ′ ) ≤ λ E ( x ) . (4) Ther e is no isometric action G y H o n a Hilb ert sp ac e such that the en e r gy function E : H → R attains a p ositive min i - mum. GROUPS OF POL Y NOMIAL GRO WTH 13 Pr o of. Clearly (2) = ⇒ (1). Also, (1 ) = ⇒ (4) since the energy function E is zero at a fixed p oint. (3) = ⇒ (2). Supp o se (3) holds. Let G y H b e an isometric a ctio n, and pic k x 0 ∈ H . Define a sequence { x k } ⊂ H inductiv ely , b y c ho o sing x k +1 ∈ B ( x k , D p E ( x k )) suc h that E ( x k +1 ) ≤ λE ( x k ). Then E ( x k ) ≤ λ k E ( x 0 ) and d ( x k +1 , x k ) ≤ D p E ( x k ) ≤ D λ k 2 p E ( x 0 ). Therefore { x k } is Cauc h y , with limit x ∞ satisfying d ( x ∞ , x 0 ) ≤ D p E ( x 0 ) 1 − λ 1 2 . Then E ( x ∞ ) = lim k →∞ E ( x k ) = 0, and x ∞ is fixed b y G . Therefore (2) holds. (4) = ⇒ (3). W e pro v e the contrap o sitiv e. Assum e that (3) fails. Then for ev ery k ∈ N , we can find an is ometric action G y H k on a Hilb ert space, and a p oint x k ∈ H k suc h that (A.2) E ( y ) >  1 − 1 k  E ( x k ) for ev ery y ∈ B ( x k , k p E ( x k )). Note that in particular, E ( x k ) >  1 − 1 k  E ( x k ), forcing E ( x k ) > 0. Let H ′ k b e the result of r escaling the metric on H k b y 1 √ E ( x k ) . Then (A.2) implies that the induced isometric action G y H ′ k satisfies E ( x k ) = 1 a nd (A.3) E ( y ) ≥ 1 − 1 k for all y ∈ B ( x k , k ). Then an y ultra limit (see [Gro 93, KL97]) of the sequence ( H k , x k ) of p oin ted Hilb ert spaces is a p ointed Hilb ert space ( H ω , x ω ) with an isometric action G y H ω suc h that E ( x ω ) = 1 = inf y ∈H ω E ( y ) . Therefore (4) fails.  Before pro cee ding we recall some f a cts ab out harmonic maps on graphs. Supp ose G is a lo cally finite metric graph, where all edges ha v e length 1. If f : G → H is a piecewise smo oth map to a Hilb ert space, then the followin g are equiv alen t: 14 BRUCE KLEIN ER • f is harmonic. • The D iric hlet energy of f (on any finite subgraph) is stationary with resp ect t o compactly supp orted v ariatio ns of f . • The re striction of f to e ac h edge of G has constan t de riv ativ e, and for eve ry v ertex v ∈ G , X d ( w, v )=1 ( f ( w ) − f ( v )) = 0 . Note that if G y H is a n isometric action on a Hilb ert space, then E is a smo o th con vex function, a nd its deriv ative is D E ( x )( v ) = 2 X s ∈ S h sx − x, ( D s )( v ) i − X s ∈ S h sx − x, v i ! = 2 X s ∈ S h x − s − 1 x, v i + X s ∈ S h x − sx, v i ! = 4 X s ∈ S h x − sx, v i . Therefore x ∈ H is a critical p oint of E ⇐ ⇒ x is a minim um of E (A.4) ⇐ ⇒ X s ∈ S ( x − sx ) = 0 . Therefore t he G -equiv ariant map f 0 : G → H giv en b y f 0 ( g ) := g x extends to a G -equiv a r ian t harmonic map f : Γ → H if and o nly if X s ∈ S ( f 0 ( se )) − f 0 ( e )) = X s ∈ S ( sx − x ) = 0 ⇐ ⇒ x is a minim um of E . The next result is a ve ry sp ecial case of a theorem from [Mok95, KS97]. Lemma A.5. The fol lowing ar e e quivalent: (1) G do es not have Pr op erty (T). (2) Ther e is an isometric action G y H on a H ilb ert sp ac e H and a nonc o n stant G -e q uivariant h armonic map f : Γ → H . Pr o of. (1) = ⇒ (2). If G do es not ha ve Prop erty (T), then b y Theorem A.1 there is an isometric action G y H on a Hilbert space, and a p oin t x ∈ H with E ( x ) = inf y ∈H E ( y ) > 0. Let f : Γ → H b e the G -equiv ariant map with f ( g ) = g x for ev ery g ∈ G ⊂ Γ, and GROUPS OF POL Y NOMIAL GRO WTH 15 whose r estriction to each edge e of Γ has constan t deriv a tiv e. The n f is harmonic, and obv iously nonconstant. (2) = ⇒ (1). 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