The Hattori-Stallings rank, the Euler-Poincaré characteristic and zeta functions of totally disconnected locally compact groups

Abstract

For a unimodular totally disconnected locally compact group $G$ we introduce and study an analogue of the Hattori-Stallings rank $\tilde{\rho }(P)\in {\mathbf{h}}_G$ for a finitely generated projective rational discrete left $\mathbb{Q}[G]$-module $P$. Here ${\mathbf{h}}_G$ denotes the $\mathbb{Q}$-vector space of left invariant Haar measures of $G$. Indeed, an analogue of Kaplansky's theorem holds in this context (cf. Theorem A). As in the discrete case, using this rank function it is possible to define a rational discrete Euler-Poincaré characteristic ${\tilde{\chi }}_G$ whenever $G$ is a unimodular totally disconnected locally compact group of type ${\mathrm{FP}}_{\infty }$ of finite rational discrete cohomological dimension. E.g., when $G$ is a discrete group of type $\mathrm{FP}$, then ${\tilde{\chi }}_G$ coincides with the ''classical'' Euler-Poincaré characteristic times the counting measure ${\mu }_{\{1\}}$. For a profinite group $\mathcal{O}$, ${\tilde{\chi }}_{\mathcal{O}}$ equals the probability Haar measure ${\mu }_{\mathcal{O}}$ on $\mathcal{O}$. Many more examples are calculated explicitly (cf. Example 1.7 and Section 5). In the last section, for a totally disconnected locally compact group $G$ satisfying an additional finiteness condition, we introduce and study a formal Dirichlet series ${\zeta }_{{}_{G,\mathcal{O}}}(s)$ for any compact open subgroup $\mathcal{O}$. In several cases it happens that ${\zeta }_{{}_{G,\mathcal{O}}}(s)$ defines a meromorphic function ${\tilde{\zeta }}_{{}_{G,\mathcal{O}}}:\mathbb{C}\to \bar{\mathbb{C}}$ of the complex plane satisfying miraculously the identity ${\tilde{\chi }}_G={\tilde{\zeta }}_{{}_{G,\mathcal{O}}}(-1{)}^{-1}\cdot {\mu }_{\mathcal{O}}$. Here ${\mu }_{\mathcal{O}}$ denotes the Haar measure of $G$ satisfying ${\mu }_{\mathcal{O}}(\mathcal{O})=1$.

42 pages