Structural description of dihedral extended Schottky groups and application in study of symmetries of handlebodies

TL;DR

This paper provides a geometric description of dihedral extended Schottky groups, which are generated by two extended Schottky groups with the same orientation-preserving half, and applies this to study symmetries of handlebodies.

Contribution

It introduces a geometric structural framework for dihedral extended Schottky groups and demonstrates their application in analyzing symmetries of handlebodies with Schottky structures.

Findings

Derived sharp upper bounds for the number of fixed point components of symmetries in handlebodies.

Provided a geometric description of dihedral extended Schottky groups using Klein-Maskit theorems.

Applied the structural results to study symmetries of three-dimensional manifolds.

Abstract

Given a symmetry $\tau$ of a closed Riemann surface $S$, there exists an extended Kleinian group $K$, whose orientation-preserving half is a Schottky group $\Gamma$ uniformizing $S$, such that $K/\Gamma$ induces $\langle \tau \rangle$; the group $K$ is called an extended Schottky group. A geometrical structural description, in terms of the Klein-Maskit combination theorems, of both Schottky and extended Schottky groups is well known. A dihedral extended Schottky group is a group generated by the elements of two different extended Schottky groups, both with the same orientation-preserving half. Such configuration of groups corresponds to closed Riemann surfaces together with two different symmetries and the aim of this paper is to provide a geometrical structure of them. This result can be used in study of three dimensional manifolds and as an illustration we give the sharp upper bounds for the total number of connected components of the locus of fixed points of two and three different symmetries of a handlebody with a Schottky structure.