The Hattori-Stallings rank, the Euler-Poincar\'e 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 ρ̃(P)G for a finitely generated projective rational discrete left Q[G]-module P. Here hG denotes the 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-Poincar\'e characteristic χ̃G whenever G is a unimodular totally disconnected locally compact group of type FP_∞ of finite rational discrete cohomological dimension. E.g., when G is a discrete group of type FP, then χ̃G coincides with the ''classical'' Euler-Poincar\'e characteristic times the counting measure μ_\1\. For a profinite group O, χ̃O equals the probability Haar measure μO on 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,O(s) for any compact open subgroup O. In several cases it happens that ζ_G,O(s) defines a meromorphic function ζ̃_G,O C → C̄ of the complex plane satisfying miraculously the identity χ̃G=ζ̃_G,O(-1)⁻¹·μO. Here μO denotes the Haar measure of G satisfying μO(O)=1.