Representation zeta functions of self-similar branched groups

Abstract

We compute the numbers of irreducible linear representations of self-similar branched groups, by expressing these numbers as the coëfficients $r_n$ of a Dirichlet series $\sum r_n{n}^{-s}$.

We show that this Dirichlet series has a positive abscissa of convergence and satisfies a functional equation thanks to which it can be analytically continued (through root singularities) to the right half-plane.

We compute the abscissa of convergence and the functional equation for some prominent examples of branched groups, such as the Grigorchuk and Gupta–Sidki groups.