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

Source abstract

For a unimodular totally disconnected locally compact group G G we introduce and study an analogue of the Hattori–Stallings rank ρ ~ ( P ) ∈ h G \tilde {\rho }(P)\in \mathbf {h}_G for a finitely generated projective rational discrete left Q [ G ] \mathbb {Q}[G] -module P P . Here h G \mathbf {h}_G denotes the Q \mathbb {Q} -vector space of left invariant Haar measures of G G . Moreover, an analogue of Kaplansky’s theorem holds in this context. As in the discrete case, using this rank function it is possible to define the rational discrete Euler–Poincaré characteristic χ ~ G \tilde {\chi }_G of G G . To this end, G G must satisfy three properties: unimodularity, being of type F P ∞ \mathrm {FP}_\infty and having finite rational discrete cohomological dimension. E.g., if G G is a discrete group of type F P \mathrm {FP} then χ ~ G \tilde {\chi }_G coincides with the classical Euler–Poincaré characteristic times the counting measure μ { 1 } \mu _{\{1\}} . If G G is profinite then χ ~ G \tilde {\chi }_G equals 1 ⋅ μ G 1\cdot \mu _G , where μ G \mu _G denotes the probability Haar measure on G G . Many more examples are calculated explicitly. In the last section we introduce and study a formal Dirichlet series ζ G , O ( s ) \zeta _{_{G,\mathcal {O}}}(s) for a totally disconnected locally compact group G G , satisfying an additional condition, and a compact open subgroup O \mathcal {O} of G G . In several cases ζ G , O ( s ) \zeta _{_{G,\mathcal {O}}}(s) defines a meromorphic function ζ ~ G , O : C → C ¯ \tilde {\zeta }_{_{G,\mathcal {O}}}\colon \mathbb {C}\to \bar {\mathbb {C}} of the complex plane satisfying miraculously the identity χ ~ G = ζ ~ G , O ( − 1 ) − 1 ⋅ μ O \tilde {\chi }_G=\tilde {\zeta }_{_{G,\mathcal {O}}}(-1)^{-1}\cdot \mu _\mathcal {O} . Here μ O \mu _{\mathcal {O}} denotes the Haar measure of G G satisfying μ O ( O ) = 1 \mu _{\mathcal {O}}(\mathcal {O})=1 .