Representation zeta functions of wreath products with finite groups

Abstract

Let $G$ be a group which has a finite number $r_n(G)$ of irreducible linear representations in ${\mathrm{GL}}_n(\mathbb{C})$ for all $n\ge 1$. Let $\zeta (G,s)={\sum }_{n=1}^{\infty }{r}_n(G)n^{-s}$ be its representation zeta function.

First, in case $G=H{\wr }_{X}Q$ is a permutational wreath product with respect to a permutation group $Q$ 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 Möbius function associated to the lattice ${\Pi }_Q(X)$ of partitions of $X$ in orbits under subgroups of $Q$.

Then we consider groups $W(Q,k)=(\cdots (Q{\wr }_{X}Q){\wr }_{X}Q\cdot \cdot \cdot ){\wr }_{X}Q$ which are iterated wreath products (with $k$ factors $Q$), and several related infinite groups $W(Q)$, including the profinite group ${\lim }_{\leftarrow k}W(Q,k)$, a locally finite group ${\lim }_{k}W(Q,k)$, and several finitely generated dense subgroups of ${\lim }_{\leftarrow k}W(Q,k)$. Under convenient hypotheses (in particular $Q$ should be perfect), we show that $r_n(W(Q))<\infty$ for all $n\ge 1$, and we establish that the Dirichlet series $\zeta (W(Q),s)$ has a finite and positive abscissa of convergence ${\sigma }_0={\sigma }_0(W(Q))$. Moreover, the function $\zeta (W(Q),s)$ satisfies a remarkable functional equation involving $\zeta (W(Q),es)$ for $e\in \{1,\dots ,d\}$, where $d=|X|$. As a consequence of this, we exhibit some properties of the function, in particular that $\zeta (W(Q),s)$ has a root-type singularity at ${\sigma }_0$, with a finite value at ${\sigma }_0$ and a Puiseux expansion around ${\sigma }_0$ .

We finally report some numerical computations for $Q=A_5$ and $Q={\mathrm{PGL}}_3({\mathbb{F}}_2)$.