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

Abstract

It is shown that $2β_1(\G)\leq h(\G)$ for any countable group $\G$, where $β_1(\G)$ 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 $$2{\hat{I}}_1^2(R)\le 2C(R)-2\le h(R),$$ where ${\hat{I}}_1^2(\mathbb{R})$ 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 $\hat{I}\pm$ of $\G$ of the uniform isoperimetric constant $h({\mathbb{R}}_{\hat{I}}\pm )$ of the equivalence relation $R_{\hat{I}}\pm$ associated to $\hat{I}\pm$. 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).

17 pp