# Groups, Geometry, and Dynamics

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**Volume 11, Issue 1, 2017, pp. 29–56**

**DOI: 10.4171/GGD/386**

Published online: 2017-04-20

Representation zeta functions of self-similar branched groups

Laurent Bartholdi^{[1]}(1) Universität Göttingen, Germany

We compute the numbers of irreducible linear representations of self-similar branched groups, by expressing these numbers as the coëfficients $r_n$ of a Dirichlet series $\sum r_n n^{-s}$.

We show that this Dirichlet series has a positive abscissa of convergence and satisfies a functional equation thanks to which it can be analytically continued (through root singularities) to the right half-plane.

We compute the abscissa of convergence and the functional equation for some prominent examples of branched groups, such as the Grigorchuk and Gupta–Sidki groups.

*Keywords: *Self-similar groups, representation growth, zeta function, functional equation, analytic continuation

Bartholdi Laurent: Representation zeta functions of self-similar branched groups. *Groups Geom. Dyn.* 11 (2017), 29-56. doi: 10.4171/GGD/386