Double-coset zeta functions for groups acting on trees

TL;DR

This paper investigates double-coset zeta functions for groups acting on trees, providing explicit formulas and exploring their properties, including connections to Euler characteristics and Ihara zeta functions.

Contribution

It offers a geometric characterization of convergence and explicit determinant formulas for the zeta functions in terms of local action data, advancing understanding of these functions.

Findings

Explicit determinant formulas for zeta functions.

Evaluation at -1 relates to Euler-Poincaré characteristic.

Connections established with Ihara zeta functions.

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-Poincar\'e 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.