Representation zeta functions of wreath products with finite groups

Abstract

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Let G be a group which has a finite number rn(G) of irreducible linear representations in GLn(ℂ) for all n ≥ 1. Let ζ(G,s) = ∑∞n = 1rn(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 ΠQ(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← kW(Q,k), a locally finite group limkW(Q,k), and several finitely generated dense subgroups of lim← kW(Q,k). Under convenient hypotheses (in particular Q should be perfect), we show that rn(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 σ0 = σ0(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 σ0, with a finite value at σ0 and a Puiseux expansion around σ0.

We finally report some numerical computations for Q = A5 and Q = PGL3(F2).