Thomas Weigel

\mathbf \zeta-functions associated to totally-disconnected, locally compact groups

It is a well-known technique to use formal Dirichlet series as a counting function for many combinatorial problems in Mathematics. In the lecture we will introduce and study certain formal Dirichlet series \zeta_{G,\mathcal{O}} one can associate to a totally-disconnected, locally-compact (=t.d.l.c.) group G  satisfying some weak finiteness properties and a compact open subgroup {\mathcal O}. Indeed, for many known examples of t.d.l.c. groups G, the abscissa of convergency \mathrm{abs}(\zeta_{G,\mathcal{O}}) will be finite and thus defines a holomorphic function \hat{\zeta}_{G,\mathcal{O}} of some half-plane of {\mathbb C}. For some classes of t.d.l.c. groups this function turns out to have a meromorphic continuation to the whole complex plane. In this case there is a miraculous connection between the Euler-Poincaré characteristic of G and the value \hat{\zeta}_{G,\mathcal{O}}(-1). If time permits we will also present a potential interpretation of the order of the pole at \mathrm{abs}(\zeta_{G,\mathcal{O}}) in case that \mathrm{abs}(\zeta_{G,\mathcal{O}}) equals 0.