-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 .