Variation of the Hausdorff dimension and degenerations of Schottky groups

Abstract

We show that the Hausdorff dimension of the limit set of a Schottky group varies continuously over the moduli space of Schottky groups defined over any complete valued field constructed by Poineau and Turchetti. To obtain this result, we first study the non-Archimedean case in a setting of Berkovich analytic spaces, where we make use of Poincar\'e series. We show that the latter can be extended meromorphically over the complex plane and admit a special value at zero which is a purely topological invariant. As an application, we prove results on the asymptotic behavior of the Hausdorff dimension of degenerating families of complex Schottky groups. For certain families, including Schottky reflection groups, we obtain an exact formula for the asymptotic logarithmic decay rate of the Hausdorff dimension. This generalizes a theorem of McMullen.