Uniform non-amenability, cost, and the first ${\ell }^2$-Betti number

Abstract

It is shown that 2$\beta$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$\beta$_1($R$) ≤ 2C($R$) − 2 ≤ $h$($R$),

where $\beta$_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 $h$($R_{\alpha }$) of the equivalence relation $R_{\alpha }$ associated to $\alpha$. 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).