Sharp upper bounds for the number of fixed points components of two and three symmetries of handlebodies

TL;DR

This paper establishes precise upper bounds on the total number of fixed point components for two and three symmetries of handlebodies, using the structure of extended Schottky groups.

Contribution

It provides a structural description of dihedral extended Schottky groups and derives sharp upper bounds for fixed point components of symmetries in handlebodies.

Findings

Sharp upper bounds for fixed point components of symmetries.

Structural characterization of dihedral extended Schottky groups.

Application to handlebody symmetry analysis.

Abstract

An extended Kleinian group whose orientation-preserving half is a Schottky group is called an extended Schottky group. These groups correspond to the real points in the Schottky space. Their geometric structures is well known and it permits to provide information on the locus of fixed points of symmetries of handlebodies. A group generated by two different extended Schottky groups, both with the same orientation-preserving half, is called a dihedral extended Schottky group. We provide a structural description of these type of groups and, as a consequence, we obtain sharp upper bounds for the sum of the cardinalities of the connected components of the locus of fixed points of two or three different symmetries of a handlebody.