Representation Zeta Functions of Wreath Products with Finite Groups

Let G be a group which has a finite number hn(G) of irreducible linear representations in GLn(C) for all n ≥ 1. Let ζ(G, s) = P∞ n=1 hn(G)n −s be its representation zeta function. First, in case G = H oXQ 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 ΠQ(X) of partitions of X in orbits under subgroups of Q. Then, we consider groupsW (Q, k) = (· · · (Q oX Q) oX Q · · · ) oX 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 limk 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 hn(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 s0 = s0(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 singularity at s0, a finite value at s0, and a Puiseux expansion around s0. We finally report some numerical computations for Q = A5 and Q = PGL3(F2).