## 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 ${\beta^1_{(2)}(G)=\frac{k\ell-k-\ell}{k\ell}}$.