A gentle introduction to the Euler-Poincaré characteristic of totally disconnected locally compact groups

The Euler-Poincaré characteristic of a discrete group is an important (but also quite mysterious) invariant. It is usually just an integer or a rational number and reflects many quite significant properties. The realm of totally disconnected locally compact groups admits an analogue of the Euler-Poincaré characteristic which surprisingly is no longer just an integer, or a rational number, but a rational multiple of a Haar measure. What arouses our curiosity is also the fact that – in some cases – the Euler-Poincaré characteristic turns out to be miraculously related to a zeta-function. 
Joint work with Gianmarco Chinello and Thomas Weigel.