Representation zeta functions of wreath products with finite groups

  • Laurent Bartholdi

    Georg-August-Universität Göttingen, Germany
  • Pierre de la Harpe

    Université de Genève, Switzerland


Let be a group which has a finite number of irreducible linear representations in for all . Let be its representation zeta function.

First, in case is a permutational wreath product with respect to a permutation group on a finite set , we establish a formula for in terms of the zeta functions of and of subgroups of , and of the Möbius function associated to the lattice of partitions of in orbits under subgroups of .

Then we consider groups which are iterated wreath products (with factors ), and several related infinite groups , including the profinite group , a locally finite group , and several finitely generated dense subgroups of . Under convenient hypotheses (in particular should be perfect), we show that for all , and we establish that the Dirichlet series has a finite and positive abscissa of convergence . Moreover, the function satisfies a remarkable functional equation involving for , where . As a consequence of this, we exhibit some properties of the function, in particular that has a root-type singularity at , with a finite value at and a Puiseux expansion around .

We finally report some numerical computations for and .

Cite this article

Laurent Bartholdi, Pierre de la Harpe, Representation zeta functions of wreath products with finite groups. Groups Geom. Dyn. 4 (2010), no. 2, pp. 209–249

DOI 10.4171/GGD/81