Cyclic-Schottky strata of Schottky space

Abstract: Schottky space ${\mathcal S}{g}$, where $g \geq 2$ is an integer, is a connected complex orbifold of dimension $3(g-1)$; it provides a parametrization of the ${\rm PSL}{2}({\mathbb C})$-conjugacy classes of Schottky groups $Γ$ 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 $[Γ] \in {\mathcal B}{g}$, then there is a Kleinian group $K$ containing $Γ$ as a normal subgroup of index some prime integer $p \geq 2$. 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 $t,r,s \geq 0$ 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 $p \geq 3$, we study the connectivity of these $F(g,p;t,r,s)$.