Representation zeta functions of self-similar branched groups

Abstract

We compute the number of irreducible linear representations of self-similar branch groups, by expressing these numbers as the coëfficients 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.

arXiv comment13 pages. Fixed a typo in a formula for the Gupta-Sidki group