Representation zeta functions of self-similar branched groups (Q2357400)

Summary: We compute the numbers of irreducible linear representations of self-similar branched groups, by expressing these numbers as the coefficients \(r_n\) 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.