The Hattori-Stallings rank, the Euler-Poincaré characteristic and zeta functions of totally disconnected locally compact groups
For a unimodular totally disconnected locally compact group $G$ we introduce and study an analogue of the Hattori-Stallings rank $\tildeρ(P)\in\mathbf{h}_G$ 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 $\tildeχ_G$ 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 $\tildeχ_G$ coincides with the ''classical'' Euler-Poincaré characteristic times the counting measure $μ_{\{1\}}$. For a profinite group $\mathcal{O}$, $\tildeχ_{\mathcal{O}}$ equals the probability Haar measure $μ_{\mathcal{O}}$ 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 $ζ_{_{G,\mathcal{O}}}(s)$ for any compact open subgroup $\mathcal{O}$. In several cases it happens that $ζ_{_{G,\mathcal{O}}}(s)$ defines a meromorphic function $\tildeζ_{_{G,\mathcal{O}}}\colon \mathbb{C} \to\bar{\mathbb C}$ of the complex plane satisfying miraculously the identity $\tildeχ_G=\tildeζ_{_{G,\mathcal{O}}}(-1)^{-1}\cdotμ_{\mathcal{O}}$. Here $μ_{\mathcal{O}}$ denotes the Haar measure of $G$ satisfying $μ_{\mathcal{O}}(\mathcal{O})=1$.