A new proof of Gromov's theorem on groups of polynomial growth

We give a new proof of Gromov's theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.

and has weakly polynomial growth if for some d ∈ (0, ∞) We give a proof of the following special case of a theorem of Colding-Minicozzi, without using Gromov's theorem on groups of polynomial growth: Theorem 1.4 ( [CM97]).Let Γ be a Cayley graph of a group G of weakly polynomial growth, and d ∈ [0, ∞).Then the space of harmonic functions on Γ with polynomial growth at most d is finite dimensional.
Note that although [CM97] stated the result for groups of polynomial growth, their proof also works for groups of weakly polynomial growth, in view of [vdDW84].
We then use this to derive the following corollaries: Corollary 1.5.If G is an infinite group of weakly polynomial growth, then G admits a finite dimensional linear representation G → GL(n, R) with infinite image.
We emphasize that our proof of Corollary 1.6 yields a new proof of Gromov's theorem on groups of polynomial growth, which does not involve the Montgomery-Zippin-Yamabe structure theory of locally compact groups [MZ74]; however, it still relies on Tits' alternative for linear groups [Tit72] (or the easier theorem of Shalom that amenable linear groups are virtually solvable [Sha98]).
Remark 1.7.There are several important applications of the Wilkie-Van Den Dries refinement [vdDW84] of Gromov's theorem [Gro81] that do not follow from the original statement; for instance [Pap05], or the theorem of Varopoulos that a group satisfies a d-dimensional Euclidean isoperimetric inequality unless it is virtually nilpotent of growth exponent < d.
1.2.Sketch of the proofs.The proof of Theorem 1.4 is based on a new Poincare inequality which holds for any Cayley graph Γ of any finitely generated group G: Here f is a piecewise smooth function on B(3R), f R is the average of u over the ball B(R), and S is the generating set for G.
The remainder of the proof has the same rough outline as [CM97], though the details are different.Note that [CM97] assumes a uniform doubling condition as well as a uniform Poincare inequality.In our context, we may not appeal to such uniform bounds as their proof depends on Gromov's theorem.Instead, the idea is to use (1.8) to show that one has uniform bounds at certain scales, and that this is sufficient to deduce that the space of harmonic functions in question is finite dimensional.
The proof of Corollary 1.5 invokes a Theorem of [Mok95,KS97] to produce a fixed point free isometric G-action G H, where H is a Hilbert space, and a G-equivariant harmonic map f : Γ → H from the Cayley graph of G to H. Theorem 1.4 then implies that f takes values in a finite dimensional subspace of H, and this implies Corollary 1.5.See Section 4.
Corollary 1.6 follows from Corollary 1.5 by induction on the degree of growth, as in the original proof of Gromov; see Section 5. 1.3.Acknowledgements.I would like to thank Alain Valette for an inspiring lecture at MSRI in August 2007, and the discussion afterward.This gave me the initial impetus to find a new proof of Gromov's theorem.I would especially like to thank Laurent Saloff-Coste for telling me about the Poincare inequality in Theorem 2.2, which has replaced a more complicated one used in an earlier draft of this paper, and Bill Minicozzi for simplifying Section 3. Finally I want to thank Toby Colding for several conversations regarding [CM97], and Emmaneul Breuillard, David Fisher, Misha Kapovich, Bill Minicozzi, Lior Silberman and Alain Valette for comments and corrections.

A Poincare inequality for finitely generated groups
Let G be a group, with a finite generating set S ⊂ G.We denote the associated word norm of g ∈ G by |g|.
We will denote the R-ball in the associated Cayley graph by B(R) = B(e, R).
Remark 2.1.We are viewing the Cayley graph as (the geometric realization of a) 1-dimensional simplicial complex, not as a discrete space.Thus B G (R) is a finite set, whereas B(R) is typically 1-dimensional.
Theorem 2.2.For every R ∈ [0, ∞) ∩ Z and every smooth function where f R is the average of f over B(R).
For every y ∈ G, we choose a shortest vertex path For every ordered pair (e 1 , e 2 ) of edges contained in B(R), let x i ∈ e i ∩ G be elements such that d(x 1 , x 2 ) ≤ 2R − 2, and let y = x −1 1 x 2 .By the Cauchy-Schwarz inequality, (2.5) where x 1 and y are as defined above.The map (e 1 , e 2 ) → (x 1 , y) is at most |S| 2 -to-one, so Remark 2.6.Although the theorem above is not in the literature, the proof is virtually contained in [CSC93,.When hearing of my more complicated Poincare inequality, Laurent Saloff-Coste's immediate response was to state and prove Theorem 2.2.
3. The proof of Theorem 1.4 In this section G will be a finitely generated group with a fixed finite generating set S, and the associated Cayley graph and word norm will be denoted Γ and • , respectively.For R ∈ Z + we let Let V be a 2k-dimensional vector space of harmonic functions on Γ.We equip V with the family of quadratic forms {Q R } R∈[0,∞) , where The remainder of this section is devoted to proving the following statement, which clearly implies Theorem 1.4: The overall structure of the proof is similar to that of Colding-Minicozzi [CM97].
3.1.Finding good scales.We begin by using the polynomial growth assumption to select a pair of comparable scales R 1 < R 2 at which both the growth function V and the determinant (det Q R ) 1 dim V have doubling behavior.Later we will use this to find many functions in V which have doubling behavior at scale R 2 .Similar scale selection arguments appear in both [Gro81] and [CM97]; the one here is a hybrid of the two.

Observe that the family of quadratic forms
We define f : Note that since Q R is a nondecreasing function of R, both f and h are nondecreasing functions, and (3.2) translates to Put a = 4d log 16, and pick w ∈ N.
Otherwise, for all l ∈ N we would get which contradicts (3.3) for large l.
These i 1 and i 2 satisfy the conditions of the lemma, because and let B j := B(x j , R 1 ).Then the collection B := {B j } j∈J covers B(R 2 ), and 1 2 B := { 1 2 B j } j∈J is a disjoint collection.Lemma 3.9.
(3) There is a C ∈ (0, ∞) depending only on |S| such that for every j ∈ J and every smooth function v : 3B i → R, (3.10) Proof.
(3) By Theorem 2.2 and the translation invariance of the inequality, 3.3.Estimating functions relative to the cover B. We now estimate the size of a harmonic function in terms of its averages over the B j 's, and its size on a larger ball.
We define a linear map Φ : Lemma 3.11 (cf.[CM97, Prop.2.5]).There is a constant C ∈ (0, ∞) depending only on the size of the generating set S, with the following property.
(1) If u is a smooth functions on B(16R 2 ), then (2) If u is harmonic on B(16R 2 ), then Proof.We will use C to denote a constant which depends only on |S|; however, its value may vary from equation to equation.
We have We estimate each of the terms in (3.14) in turn.
Inequality (3.13) follows from (3.12) by applying the reverse Poincare inequality, which holds for any harmonic function v defined on B(16R 2 ): (For the proof, see [SY95, Lemma 6.3], and note that for harmonic functions their condition u ≥ 0 may be dropped.) 3.4.Selecting functions from V with controlled growth.Our next step is to select functions in V which have doubling behavior at scale R 2 .
3.5.Bounding the dimension of V. We now assume that w is the smallest integer such that where C is the constant in (3.13).Therefore 2 • 16 −(w−1) ≥ 1 2Ce 4a , and this implies e wa ≤ 64 C e 64d 2 log 16 .
4. Proof of Corollary 1.5 using Theorem 1.4 Let G be as in the statement of the Corollary, and let Γ denote some Cayley graph of G with respect to a symmetric finite generating set S.
k for all k, then for every k there must be an k ; this means that the sequence of balls {B G (r k )} is a Folner sequence for G.
Hence G does not have Property (T).Therefore by a result of Mok [Mok95] and Korevaar-Schoen [KS97, Theorem 4.1.2],there is an isometric action G H of G on a Hilbert space H which has no fixed points, and a nonconstant G-equivariant harmonic map f : Γ → H.
In the case of Cayley graphs, the Mok/Korevaar-Schoen result is quite elementary, so we give a short proof in Appendix A.
Since f is G-equivariant, it is Lipschitz.
Each bounded linear functional φ ∈ H * gives rise to a Lipschitz harmonic function φ • f , and hence we have a linear map Φ : H * → V, where V is the space of Lipschitz harmonic functions on Γ.Since the target is finite dimensional by Theorem 1.4, the kernel of Φ has finite codimension, and its annihilator ker(Φ) ⊥ ⊂ H is a finite dimensional subspace containing the image of f .It follows that the affine hull A of the image of f is finite dimensional and G-invariant.Therefore we have an induced isometric G-action G A. This action cannot factor through a finite group, because it would then have fixed points, contradicting the fact that the original representation is fixed point free.The associated homomorphism G → Isom(A) yields the desired finite dimensional representation of G.

Proof of Corollary 1.6 using Corollary 1.5
We prove Gromov's theorem using Corollary 1.5.The proof is a recapitulation of Gromov's argument, which reproduce here for the convenience of the reader.
The proof is by induction on the degree of growth.A group whose degree of growth is 0 is finite, and hence Corollary 1.6 holds for such a group.Assume inductively that for some d ∈ N that every group of degree at most d − 1 is virtually nilpotent, and suppose deg(G) = d.Then G is infinite, and by Corollary 1.5 there is a finite dimensional linear representation G → GL(n) with infinite image H ⊂ GL(n).Since H has polynomial growth, by [Tit72] (see [Sha98] for an easier proof) it is virtually solvable, and by [Wol68,Mil68] it must be virtually nilpotent.
After passing to finite index subgroups, we may assume H is nilpotent, and that its abelianization is torsion-free.It follows that there is • f is harmonic.
• The Dirichlet energy of f (on any finite subgraph) is stationary with respect to compactly supported variations of f .• The restriction of f to each edge of G has constant derivative, and for every vertex v ∈ G, (f (w) − f (v)) = 0.

Note that if G
H is an isometric action on a Hilbert space, then E is a smooth convex function, and its derivative is (1) G does not have Property (T).
(2) There is an isometric action G H on a Hilbert space H and a nonconstant G-equivariant harmonic map f : Γ → H.
Proof.(1) =⇒ (2).If G does not have Property (T), then by Theorem A.1 there is an isometric action G H on a Hilbert space, and a point x ∈ H with E(x) = inf y∈H E(y) > 0. Let f : Γ → H be the G-equivariant map with f (g) = gx for every g ∈ G ⊂ Γ, and whose restriction to each edge e of Γ has constant derivative.Then f is harmonic, and obviously nonconstant.
(2) =⇒ (1).Suppose (2) holds, and f : Γ → H is the G-equivariant harmonic map.Then f (e) is a positive minimum of E : H → R; in particular the action G H has no fixed points.Therefore G does not have Property (T).
Statement of results.Definition 1.1.Let G be a finitely generated group, and let B G (r) ⊂ G denote the ball centered at e ∈ G with respect to some fixed word norm on G.The group G has polynomial growth if for some d ∈ (0, ∞) February 2, 2008.Supported by NSF Grant DMS 0701515. 1 Definition 5.1.Let G be a finitely generated group.The degree (of growth) of G is the minimum deg(G) of the nonnegative integers d such that lim inf r→∞ V (r) r d < ∞.
DE(x)(v) = 2 s∈S sx − x, (Ds)(v) − s∈S sx − x, v = 2 s∈S x − s −1 x, v + s∈S x − sx, v = 4 s∈S x − sx, v .Therefore x ∈ H is a critical point of E ⇐⇒ x is a minimum of E (A.4) ⇐⇒ s∈S (x − sx) = 0.Therefore the G-equivariant map f 0 : G → H given by f 0 (g) := gx extends to a G-equivariant harmonic map f : Γ → H if and only ifs∈S (f 0 (se)) − f 0 (e)) = s∈S (sx − x) = 0 ⇐⇒ x is a minimum of E.The next result is a very special case of a theorem from [Mok95, KS97].Lemma A.5.The following are equivalent: