Notes of Alain Valette’s Rennes lecture nr 1

{L^2}-Betti numbers for locally compact groups

1. {L^2}-Betti numbers

I missed the beginning of Valette’s minicourse.

2. Measurable equivalence relations

Gaboriau defines {L^2}-Betti numbers atached to a probability preserving equivalence relation with countable equivalence classes. If {\Gamma} acts freely , then the {L^2}-Betti numbers of the corresponding equivalence relation give back the {L^2}-Betti

It follows that for lattices in a locally compact group {G}, {L^2}-Betti numbers are proportional to covolume. It suggests that one can make sens of {L^2}-Betti numbers of {G}, at least when {G} is unimodular.

This was carried out by Kyed-Petersen-Vaes. Indeed, the von Neumann algebra {LG} has a semi-finite trace {\mathrm{Trace}(x^*x)=\int_G |f(g)|^2 \,dg} for {x=f} acting in the left regular representation. They define {\beta^n_{(2)}(G)} as the von-Neumann dimension of {n}-th cohomology of {G} with values in {L^2(G)}.

Compact generation implies that {\beta^1_{(2)}(G)} is finite. For {\Gamma\subset G} of finite covolume, {\beta^n_{(2)}(G)=\beta^n_{(2)}(\Gamma)/covol(\Gamma)}.

Amenability kills {L^2}-Betti numbers.

The proof involves defining {L^2}-Betti numbers for ergodic probability measure preserving actions of {G}.

Example. {G=SL(2,{\mathbb R})}. Using lattices, one gets {\beta^1_{(2)}(G)=1/2\pi}.

Example. {G=} automorphism group of a {(k,\ell)}-biregular tree. Using covolume 1 lattice {{\mathbb Z}/k{\mathbb Z}\star{\mathbb Z}/\ell{\mathbb Z}} and Cheeger-Gromov’s formula for {L^2}-Betti numbers of free products, one gets {\beta^1_{(2)}(G)=\frac{k\ell-k-\ell}{k\ell}}.


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