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An extended Kleinian group whose orientation-preserving half is a Schottkygroup is called an extended Schottky group. These groups correspond to the realpoints in the Schottky space. Their geometric structures is well known and itpermits to provide information on the locus of fixed points of symmetries ofhandlebodies. A group generated by two different extended Schottky groups, bothwith the same orientation-preserving half, is called a dihedral extendedSchottky group. We provide a structural description of these type of groupsand, as a consequence, we obtain sharp upper bounds for the sum of thecardinalities of the connected components of the locus of fixed points of twoor three different symmetries of a handlebody.

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