Year: 2010
Type: article
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 \geq 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 ···) \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_kW(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 \geq 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) .
Source: Groups Geometry and Dynamics
DOI: 10.4171/ggd/81
Authors: Laurent Bartholdi, Pierre de la Harpe
Institutions: University of Göttingen, University of Geneva
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2152757713