Representation zeta functions of wreath products with finite groups

TL;DR

This paper derives formulas for the representation zeta functions of wreath products with finite groups, analyzes their properties, and explores their behavior for various infinite and finite groups, including numerical examples.

Contribution

It provides a new explicit formula for the zeta functions of permutational wreath products and studies their analytic properties for infinite groups under certain conditions.

Findings

Derived a formula for $\

zeta(G,s)$ in terms of subgroup zeta functions and partition lattices.

Proved finiteness of representation counts and established the abscissa of convergence for certain infinite wreath product groups.

Abstract

Let G be a group which has for all n a finite number r_n(G) of irreducible complex linear representations of dimension n. Let $\zeta(G,s) = \sum_{n=1}^{\infty} r_n(G) n^{-s}$ be its representation zeta function. First, in case G is a permutational wreath product of H with a permutation group Q acting on a finite set X, we establish a formula for $\zeta(G,s)$ in terms of the zeta functions of H and of subgroups of Q, and of the Moebius function associated with the lattice of partitions of X in orbits under subgroups of Q. Then, we consider groups W(Q,k) which are k-fold iterated wreath products of Q, and several related infinite groups W(Q), including the profinite group, a locally finite group, and several finitely generated groups, which are all isomorphic to a wreath product of themselves with Q. Under convenient hypotheses (in particular Q should be perfect), we show that r_n(W(Q)) is finite for all n, and we establish that the Dirichlet series $\zeta(W(Q),s)$ has a finite and positive abscissa of convergence s_0. Moreover, the function $\zeta(W(Q),s)$ satisfies a remarkable functional equation involving $\zeta(W(Q),es)$ for e=1,...,|X|. As a consequence of this, we exhibit some properties of the function, in particular that $\zeta(W(Q),s)$ has a singularity at s_0, a finite value at s_0, and a Puiseux expansion around s_0. We finally report some numerical computations for Q the simple groups of order 60 and 168.