Uniform non-amenability, cost, and the first ℓ²-Betti number

Abstract

It is shown that 2β1(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 II1, the uniform isoperimetric constant h(R) of R is invariant under orbit equivalence and satisfies

2β1(R) ≤ 2C(R) − 2 ≤ h(R),

where β1(R) is the first ℓ2-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 II1 always contain non-amenable subtreeings.
An ergodic version he(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(Rα) of the equivalence relation Rα associated to α. By establishing a connection with the cost of measure-preserving equivalence relations, we prove that he(Γ) = 0 for any lattice Γ in a semi-simple Lie group of real rank at least 2 (while he(Γ) does not vanish in general).