Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.4257

Isoperimetric profile and random walks on locally compact solvable groups

Romain Tessera
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study a large class of amenable locally compact groups containing all solvable algebraic groups over a local field and their discrete subgroups. We show that the isoperimetric profile of these groups is in some sense optimal among amenable groups. We use this fact to compute the probability of return of symmetric random walks, and to derive various other geometric properties which are likely to be only satisfied by these groups.

Explore related subjects

math.GRmath.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Romain Tessera. 2010-12-04. Isoperimetric profile and random walks on locally compact solvable groups. https://arxiv.org/abs/0706.4257

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Random Burnside Groups

Fix integers $m\ge2$ and $k\ge1$. For every sufficiently large fixed odd exponent $M$, choose $k$ independent elements uniformly from the sphere of radius $\ell$ in the free Burnside group $B_m(M)$. We prove that the quotient by their normal closure is infinite with probability at least $1-e^{-c\ell}$. More precisely, the quotient map has no nontrivial kernel element of length at most $κ\ell$, for constants $c,κ>0$. We also consider the analogue of the Gromov positive density model for the Burnside variety.

math.GR↗
arXiv2026

Spherical growth of reciprocal classes in the Hecke Groups

Let $r \in \mathbb{Z}$, $r>2$, and let $Γ_{2r} \cong (\mathbb{Z}/2\mathbb{Z}) \ast (\mathbb{Z}/2r\mathbb{Z})$ be the Hecke group. Let $\mathcal{N}_\ell$ denote the set of conjugacy classes of reciprocal elements with cyclically reduced word length exactly $\ell$ in the standard generating set of $Γ_{2r}$. Let $P(x)$ be the polynomial \[ P(x) = x^{r+1} - 2\sum_{i=1}^{r-1} x^{i} - 1. \] Let $ρ$ be the unique positive root of $P$ and $C$ be a constant depending on $r.$ We prove that, as $\ell \to \infty$, \[|\mathcal{N}_\ell| \sim Cρ^{\left\lfloor \tfrac{\ell+1}{2} \right\rfloor}. \] Our approach exploits the free product structure of $Γ_{2r}$, together with a combinatorial counting function and recurrence relations arising from cyclically reduced representatives. We further show that the asymptotic growth rate of primitive reciprocal conjugacy classes of word length $\ell$ coincides with that of $\mathcal{N}_\ell$. This result extends previous work for odd values of $p$ and provides a uniform asymptotic bound for all Hecke groups.

math.GR↗