Cyclic‐Schottky strata of Schottky space.

Schottky space Sg${\mathcal {S}}_{g}$, where g⩾2$g \geqslant 2$ is an integer, is a connected complex orbifold of dimension 3(g−1)$3(g-1)$; it provides a parametrization of the PSL2(C)${\rm PSL}_{2}({\mathbb {C}})$‐conjugacy classes of Schottky groups Γ$\Gamma$ of rank g$g$. The branch locus Bg⊂Sg${\mathcal {B}}_{g} \subset {\mathcal {S}}_{g}$, 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]∈Bg$[\Gamma] \in {\mathcal {B}}_{g}$, then there is a Kleinian group K$K$ containing Γ$\Gamma$ as a normal subgroup of index some prime integer p⩾2$p \geqslant 2$. The structural description, in terms of Klein–Maskit Combination Theorems, of such a group K$K$ is completely determined by a triple (t,r,s)$(t,r,s)$, where t,r,s⩾0$t,r,s \geqslant 0$ are integers such that g=p(t+r+s−1)+1−r$g=p(t+r+s-1)+1-r$. For each such tuple (g,p;t,r,s)$(g,p;t,r,s)$, there is a corresponding cyclic‐Schottky stratum F(g,p;t,r,s)⊂Bg$F(g,p;t,r,s) \subset {\mathcal {B}}_{g}$. It is known that F(g,2;t,r,s)$F(g,2;t,r,s)$ is connected. In this paper, for p⩾3$p \geqslant 3$, we study the connectivity of these F(g,p;t,r,s)$F(g,p;t,r,s)$. [ABSTRACT FROM AUTHOR]