# Representation zeta functions of wreath products with finite groups

### Laurent Bartholdi

Georg-August-Universität Göttingen, Germany### Pierre de la Harpe

Université de Genève, Switzerland

## Abstract

Let $G$ be a group which has a finite number $r_{n}(G)$ of irreducible linear representations in $GL_{n}(C)$ for all $n≥1$. Let $ζ(G,s)=∑_{n =1}r_{n}(G)n_{−s}$ be its representation zeta function.

First, in case $G=H≀_{X}Q$ is a permutational wreath product with respect to a permutation group $Q$ on a finite set $X$, we establish a formula for $ζ(G,s)$ in terms of the zeta functions of $H$ and of subgroups of $Q$, and of the Möbius function associated to the lattice $Π_{Q}(X)$ of partitions of $X$ in orbits under subgroups of $Q$.

Then we consider groups $W(Q,k)=(⋯(Q≀_{X}Q)≀_{X}Q⋅⋅⋅)≀_{X}Q$ which are iterated wreath products (with $k$ factors $Q$), and several related infinite groups $W(Q)$, including the profinite group $lim_{← k}W(Q,k)$, a locally finite group $lim_{k}W(Q,k)$, and several finitely generated dense subgroups of $lim_{← k}W(Q,k)$. Under convenient hypotheses (in particular $Q$ should be perfect), we show that $r_{n}(W(Q))<∞$ for all $n≥1$, and we establish that the Dirichlet series $ζ(W(Q),s)$ has a finite and positive abscissa of convergence $σ_{0}=σ_{0}(W(Q))$. Moreover, the function $ζ(W(Q),s)$ satisfies a remarkable functional equation involving $ζ(W(Q),es)$ for $e∈{1,…,d}$, where $d=∣X∣$. As a consequence of this, we exhibit some properties of the function, in particular that $ζ(W(Q),s)$ has a root-type singularity at $σ_{0}$, with a finite value at $σ_{0}$ and a Puiseux expansion around $σ_{0}$ .

We finally report some numerical computations for $Q=A_{5}$ and $Q=PGL_{3}(F_{2})$.

## Cite this article

Laurent Bartholdi, Pierre de la Harpe, Representation zeta functions of wreath products with finite groups. Groups Geom. Dyn. 4 (2010), no. 2, pp. 209–249

DOI 10.4171/GGD/81