** -Betti numbers for locally compact groups **

**1. -Betti numbers **

I missed the beginning of Valette’s minicourse.

**2. Measurable equivalence relations **

Gaboriau defines -Betti numbers atached to a probability preserving equivalence relation with countable equivalence classes. If acts freely , then the -Betti numbers of the corresponding equivalence relation give back the -Betti

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

This was carried out by Kyed-Petersen-Vaes. Indeed, the von Neumann algebra has a semi-finite trace for acting in the left regular representation. They define as the von-Neumann dimension of -th cohomology of with values in .

Compact generation implies that is finite. For of finite covolume, .

Amenability kills -Betti numbers.

The proof involves defining -Betti numbers for ergodic probability measure preserving actions of .

**Example**. . Using lattices, one gets .

**Example**. automorphism group of a -biregular tree. Using covolume 1 lattice and Cheeger-Gromov’s formula for -Betti numbers of free products, one gets .