Representation zeta functions of wreath products with finite groups

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 $ζ(G,s) = \sum_{n=1}^{\infty} r_n(G) n^{-s}$ 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 $ζ(G,s)$ 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 $ζ(W(Q),s)$ has a finite and positive abscissa of convergence s_0. Moreover, the function $ζ(W(Q),s)$ satisfies a remarkable functional equation involving $ζ(W(Q),es)$ for e=1,...,|X|. As a consequence of this, we exhibit some properties of the function, in particular that $ζ(W(Q),s)$ 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.

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