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

Abstract

It is shown that $2\beta_1(\G)\leq h(\G)$ for any countable group $\G$, where $\beta_1(\G)$ is the first $\ell^2$-Betti number and $h(\G)$ the uniform isoperimetric constant. In particular, a countable group with non-vanishing first $\ell^2$-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 $\IIi$, the uniform isoperimetric constant $h(R)$ of $R$ is invariant under orbit equivalence and satisfies $$ 2\beta_1(R)\leq 2C(R)-2\leq h(R), $$ where $\beta_1(\R)$ is the first $\ell^2$-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 $\IIi$ always contain non-amenable subtreeings. An ergodic version $h_e(\G)$ of the uniform isoperimetric constant $h(\G)$ is defined as the infimum over all essentially free ergodic and measure preserving actions $\alpha$ of $\G$ 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 $h_e(\G)=0$ for any lattice $\G$ in a semi-simple Lie group of real rank at least 2 (while $h_e(\G)$ does not vanish in general).