Geometry of locally compact groups of polynomial growth and shape of large balls

Abstract

Abstract. We show that any locally compact group G with polynomial growth is weakly commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We then study the shape of large balls and show, generalizing work of P. Pansu, that after a suitable renormalization, they converge to a limiting compact set, which is isometric to the unit ball for a left-invariant subFinsler metric on the so-called graded nilshadow of S. As by-products, we obtain asymptotics for the volume of large balls, we prove that balls are Folner and hence that the ergodic theorem holds for all ball averages. Along the way we also answer negatively a question of Burago and Margulis [7] on asymptotic word metrics and recover some results of Stoll [33] of the rationality of growth series of Heisenberg groups. 1. Introduction 1.1. Groups with polynomial growth. Let G be a locally compact group with left Haar measure vol G . We will assume that G is generated by a compact symmetric subset Ω. Classically, G is said to have polynomial growth if there exist C > 0 and k > 0 such that for any integer n ≥ 1 ContentsDate: April 2012. where Ω n = Ω · . . . · Ω is the n-fold product set. Another choice for Ω would only change the constant C, but not the polynomial nature of the bound. One of the consequences of the analysis carried out in this paper is the following theorem:Theorem 1.1 (Volume asymptotics). Let G be a locally compact group with polynomial growth and Ω a compact symmetric generating subset of G. Then there exists c(Ω) > 0 and an integer d(G) ≥ 0 depending on G only such that the following holds:This extends the main result of Pansu [27]. The integer d(G) coincides with the exponent of growth of a naturally associated graded nilpotent Lie group, the asymptotic cone of G, and is given by the Bass-Guivarc'h formula (4) below. The constant c(Ω) will be interpreted as the volume of the unit ball of a subRiemannian Finsler metric on this nilpotent Lie group. Theorem 1.1 is a byproduct of our study of the asymptotic behavior of periodic pseudodistances on G, that is pseudodistances that are invariant under a co-compact subgroup of G and satisfy a weak kind of the existence of geodesics axiom (see Definition 4.1).Our first task is to get a better understanding of the structure of locally compact groups of polynomial growth. Guivarc'h [21] proved that locally compact groups of polynomial growth are amenable and unimodular and that every compactly generated 1 closed subgroup also has polynomial growth. Guivarc'h [21] and Jenkins [15] also characterized connected Lie groups with polynomial growth: a connected Lie group has polynomial growth if and only if it is of type (R), that is if for all x ∈ Lie(S), ad(x) has only purely imaginary eigenvalues. Such groups are solvable-by-compact and any connected nilpotent Lie group is of type (R).It is much more difficult to characterize discrete groups with polynomial growth, and this was done in a celebrated paper of Gromov [17], proving that they are virtually nilpotent. Losert [24] gen...

Polynomial growth harmonic functions on groups of polynomial volume growth

“…Besides Bass' volume growth estimate (1.1), Pansu [42] proved a more delicate volume growth property for groups of polynomial volume growth (see also Breuillard [5]). He showed that for the Cayley graph (G, S) of a group G of polynomial volume growth the following limit exists 6) where D ∈ N. Then it is easy to see (Lemma 2.1) that for any θ ≪ 1, there exists R 0 (θ, S) such that…”

Asymptotic shapes for ergodic families of metrics on Nilpotent groups

“…This follows from the Baker-Campbell-Hausdorff formula (cf. §3.3 and the proof of Lemma 5.5 in [6]). …”

The geometry of spheres in free abelian groups

“…For instance, we can say that a group element w ∈ Z 2 has a simple spelling if w = av i + bv i+1 for consecutive significant generators. We can verify that every word has a simple spelling with respect to the standard generators, whereas only 1 in 36 elements has a simple spelling in S = ±{6e 1 , e 1 , 6e 2 This does depend on the generating set-only on the convex hull, as usual, but not only on its area-and it holds uniformly at large word-lengths n, as well as when averaging over words of length ≤ n. As a check, recall that Pick's theorem says that A = i + b/2 − 1. We know that r ≤ b and i ≥ 1, which means r/2A ≤ 1, which is required for plausibility.…”

“…As a consequence, the spheres in the word metric, once normalized, converge to a limit shape. This is a small special case of the theory for finitely-generated nilpotent groups and, more generally, lattices in Lie groups of polynomial growth (see Pansu and Breuillard [9,2]). Here we give an elementary proof in terms of the combinatorial group theory and Euclidean geometry.…”