The Euler characteristic and the zeta-functions of a totally disconnected locally compact group

Abstract: The Euler characteristic and the zeta-functions of a totally disconnected locally compact group Abstract: 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. Warning: in order to gain such an invariant the group has to be unimodular and satisfy some cohomological finiteness conditions. Examples of groups satisfying these additional conditions are the fundamental groups of finite trees of profinite groups. What arouses our curiosity is the fact that - in some cases - the Euler-Poincaré characteristic turns out to be miraculously related to a zeta-function. A large part of the talk will be devoted to the introduction of the just-cited objects. We aim at concluding the presentation by facing the concrete example of the group of F-points of a split semisimple simply connected algebraic group G over F (where F denotes a non-archimedean locally compact field of residue characteristic p). Joint work with Gianmarco Chinello and Thomas Weigel.

group theory

Audience: researchers in the discipline

Symmetry in Newcastle