It is shown that 21(Γ) ≤ (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 of type II_1, the uniform isoperimetric constant () of is invariant under orbit equivalence and satisfies
2_1() ≤ 2C() − 2 ≤ (),
where _1() is the first ℓ^2-Betti number and C() the cost of in the sense of Levitt (in particular () 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 (Γ) of the uniform isoperimetric constant (Γ) is defined as the infimum over all essentially free ergodic and measure preserving actions α of Γ of the uniform isoperimetric constant () of the equivalence relation associated to . By establishing a connection with the cost of measure-preserving equivalence relations, we prove that (Γ) = 0 for any lattice Γ in a semi-simple Lie group of real rank at least 2 (while (Γ) does not vanish in general).
Cite this article
Russell Lyons, Mikaël Pichot, Stéphane Vassout, Uniform non-amenability, cost, and the first -Betti number. Groups Geom. Dyn. 2 (2008), no. 4, pp. 595–617DOI 10.4171/GGD/49