Representation zeta functions of self-similar branched groups

TL;DR

This paper analyzes the representation zeta functions of self-similar branch groups, providing explicit computations and properties of their Dirichlet series, including convergence, algebraicity, and analytic continuation, with applications to notable groups.

Contribution

It introduces a method to compute and analyze the representation zeta functions of self-similar branch groups, revealing their algebraic and analytic properties.

Findings

Dirichlet series for representation counts has positive abscissa of convergence.

Series is algebraic over a specific ring involving powers of n.

Explicitly computed abscissa and functional equations for Grigorchuk and Gupta-Sidki groups.

Abstract

We compute the number of irreducible linear representations of self-similar branch groups, by expressing these numbers as the co\"efficients a_n of a Dirichlet series sum a_n n^{-s}. We show that this Dirichlet series has a positive abscissa of convergence, is algebraic over the ring Q[2^{-s},...,P^{-s}] for some integer P, and show that it can be analytically continued (through root singularities) to the left half-plane. We compute the abscissa of convergence and the functional equation for some prominent examples of branch groups, such as the Grigorchuk and Gupta-Sidki groups.