# Groups, Geometry, and Dynamics

Full-Text PDF (206 KB) | Metadata | Table of Contents | GGD summary

**Volume 2, Issue 4, 2008, pp. 595–617**

**DOI: 10.4171/GGD/49**

Published online: 2008-12-31

Uniform non-amenability, cost, and the first ℓ^{2}-Betti number

^{[1]}, Mikaël Pichot

^{[2]}and Stéphane Vassout

^{[3]}(1) Indiana University, Bloomington, United States

(2) IHES, Bures-Sur-Yvette, France

(3) Institut de Mathématiques de Jussieu - Paris Rive Gauche, France

It is shown that 2`β`_{1}(Γ) ≤ `h`(Γ) for any countable group Γ,
where `β`_{1}(Γ) is the first ℓ^{2}-Betti number and `h`(Γ) the
uniform isoperimetric constant. In particular, a countable group with non-vanishing
first ℓ^{2}-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

`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).

*Keywords: *ℓ^{2}-Betti numbers, uniform non-amenability, measured equivalence relations

Lyons Russell, Pichot Mikaël, Vassout Stéphane: Uniform non-amenability, cost, and the first ℓ^{2}-Betti number. *Groups Geom. Dyn.* 2 (2008), 595-617. doi: 10.4171/GGD/49