Double-coset zeta functions for groups acting on trees

Abstract

We study the double-coset zeta functions for groups acting on trees, focusing mainly on weakly locally $\infty$-transitive or (P)-closed actions. After giving a geometric characterisation of convergence for the defining series, we provide explicit determinant formulae for the relevant zeta functions in terms of local data of the action. Moreover, we prove that evaluation at $-1$ satisfies the expected identity with the Euler-Poincaré characteristic of the group. The behaviour at $-1$ also sheds light on a connection with the Ihara zeta function of a weighted graph introduced by A. Deitmar.

Final version, with minor modifications according to the referees' suggestions. Fixed inaccuracies in the proofs of Prop. 3.11 and in Thm. 5.9; added some references, remarks and examples in the introduction and in Sect. 3.5; uniformised some notation according to the literature. 63 pages. To appear in J. Algebra