A new proof of Gromovs theorem on groups of polynomial growth

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.

Corollary 1.6 ( [Gro81,vdDW84]). If G is a group with weakly polynomial growth, then G is virtually nilpotent.

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. (1.8)

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.

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).

Let δf : B G (3R -1) → R be given by

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.

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 l