Cyclic-Schottky strata of Schottky space

TL;DR

This paper investigates the connectivity properties of cyclic-Schottky strata within Schottky space, focusing on cases where the index p of the normal subgroup is at least 3.

Contribution

It extends the understanding of the structure and connectivity of cyclic-Schottky strata for higher index p in Schottky space.

Findings

Connectedness of F(g,2;t,r,s) is known.

The paper studies the connectivity of F(g,p;t,r,s) for p ≥ 3.

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 $\Gamma$ 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 $p \geq 2$. 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 $F(g,p;t,r,s)$.