Representation zeta functions of wreath products with finite groups

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 zeta(G, s) = Sigma(infinity)(n=1) r(n)(G)n(-s) be its representation zeta function. First, in case G = H (sic)(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 Mobius 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) = (...(Q (sic)(X) Q) (sic)(X) Q...) (sic)(X) Q which are iterated wreath products (with k factors Q), and several related infinite groups W(Q), including the profinite group (lim) under left arrow (k) W(Q, k), a locally finite group lim(k) W(Q, k), and several finitely generated dense subgroups of (lim) under left arrow (k) W(Q, k). Under convenient hypotheses (in particular Q should be perfect), we show that r(n)(W(Q)) < infinity for all n >= 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 is an element of {1,...,d}, where d = vertical bar X vertical bar. 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 = PGL(3) (F(2)).