Finiteness properties for locally compact groups and the Euler-Poincaré characteristic

Finiteness properties of groups provide various generalisations of the properties of being "finitely generated" and "finitely presented". I will give a brief overview of the different types of finiteness properties for (abstract) groups, and then focus on locally compact groups. The aim is to introduce an Euler-Poincaré characteristic for unimodular totally disconnected locally compact (= t.d.l.c.) groups of type FP over the rationals. In this context the characteristic is no longer just a number, but a rational multiple of a Haar measure. In many cases it happens that we can define a meromorphic function of the complex plane whose value in -1 detects, miraculously, the Euler-Poincaré characteristic of the unimodular t.d.l.c. group. Moreover, I will sketch how this invariant can be used to deal with accessibility problems for t.d.l.c. groups.
Based on joint works with Chinello--Weigel and Marchionna--Weigel.