# Uniform non-amenability, cost, and the first $ℓ_{2}$-Betti number

### Russell Lyons

Indiana University, Bloomington, United States### Mikaël Pichot

IHES, Bures-Sur-Yvette, France### Stéphane Vassout

Institut de Mathématiques de Jussieu - Paris Rive Gauche, France

## Abstract

It is shown that 2$β$1(Γ) ≤ $h$(Γ) for any countable group Γ, where β1(Γ) is the first ℓ2-Betti number and h(Γ) the uniform isoperimetric constant. In particular, a countable group with non-vanishing first ℓ2-Betti number is uniformly non-amenable.

We then define isoperimetric constants in the framework of measured equivalence relations. For an ergodic measured equivalence relation $R$ of type II_1, the uniform isoperimetric constant $h$($R$) of $R$ is invariant under orbit equivalence and satisfies

2$β$_1($R$) ≤ 2C($R$) − 2 ≤ $h$($R$),

where $β$_1($R$) is the first ℓ^2-Betti number and C($R$) the cost of $R$ in the sense of Levitt (in particular $h$($R$) is a non-trivial invariant). In contrast with the group case, uniformly non-amenable measured equivalence relations of type II_1 always contain non-amenable subtreeings.

An ergodic version $h_{e}$(Γ) of the uniform isoperimetric constant $h$(Γ) is defined as the infimum over all essentially free ergodic and measure preserving actions α of Γ of the uniform isoperimetric constant $h$($R_{α}$) of the equivalence relation $R_{α}$ associated to $α$. By establishing a connection with the cost of measure-preserving equivalence relations, we prove that $h_{e}$(Γ) = 0 for any lattice Γ in a semi-simple Lie group of real rank at least 2 (while $h_{e}$(Γ) does not vanish in general).

## Cite this article

Russell Lyons, Mikaël Pichot, Stéphane Vassout, Uniform non-amenability, cost, and the first $ℓ_{2}$-Betti number. Groups Geom. Dyn. 2 (2008), no. 4, pp. 595–617

DOI 10.4171/GGD/49