Title: Symmetries of handlebodies and their fixed points: Dihedral extended Schottky groups,
Abstract: A Schottky structure on a handlebody $M$ of genus $g$ is provided by a
Schottky group of rank $g$. A symmetry (an orientation-reversing involution) of
$M$ is known to have at most $(g+1)$ connected components of fixed points. Each
of these components is either a point or a compact bordered surface (either
orientable or not) whose boundary is contained in the border of $M$. In this
paper, we derive sharp upper bounds for the total number of connected
components of the sets of fixed points of given two or three symmetries of $M$.
In order to obtain such an upper bound, we obtain a geometrical structure
description of those extended Kleinian groups $K$ containing a Schottky group
$\Gamma$ as finite index normal subgroup so that $K/\Gamma$ is a dihedral group
(called dihedral Schottky groups). Our upper bounds turn out to be different to
the corresponding ones at the level of closed Riemann surfaces. In contrast to
the case of Riemann surfaces, we observe that $M$ cannot have two different
maximal symmetries.