Cyclic-Schottky strata of Schottky space

Abstract

Schottky space ${\mathcal{S}}_g$, where $g\ge 2$ is an integer, is a connected complex orbifold of dimension $3(g-1)$; it provides a parametrization of the ${\mathrm{PSL}}_2(\mathbb{C})$-conjugacy classes of Schottky groups $\Gamma$ of rank $g$. The branch locus ${\mathcal{B}}_g\subset {\mathcal{S}}_g$, consisting of those conjugacy classes of Schottky groups being a finite index proper normal subgroup of some Kleinian group, is known to be connected. If $[\Gamma ]\in {\mathcal{B}}_g$, then there is a Kleinian group $K$ containing $\Gamma$ as a normal subgroup of index some prime integer $p\ge 2$. The structural description, in terms of Klein-Maskit Combination Theorems, of such a group $K$ is completely determined by a triple $(t,r,s)$, where $t,r,s\ge 0$ are integers such that $g=p(t+r+s-1)+1-r$. For each such a tuple $(g,p;t,r,s)$ there is a corresponding cyclic-Schottky stratum $F(g,p;t,r,s)\subset {\mathcal{B}}_g$. It is known that $F(g,2;t,r,s)$ is connected.In this paper, for $p\ge 3$, we study the connectivity of these $F(g,p;t,r,s)$.