Cyclic-Schottky strata of Schottky space
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910192732471296 |
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| author | Hidalgo, Ruben A. Izquierdo, Milagros |
| author_facet | Hidalgo, Ruben A. Izquierdo, Milagros |
| contents | 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 $F(g,p;t,r,s)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04205 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cyclic-Schottky strata of Schottky space Hidalgo, Ruben A. Izquierdo, Milagros Geometric Topology 30F10, 30F40 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 $F(g,p;t,r,s)$. |
| title | Cyclic-Schottky strata of Schottky space |
| topic | Geometric Topology 30F10, 30F40 |
| url | https://arxiv.org/abs/2605.04205 |