Structure of Whittaker groups and applications to conformal involutions on handlebodies

Abstract

The geometrically finite complete hyperbolic Riemannian metrics in the interior of a handlebody of genus g, having injectivity radius bounded away from zero, are exactly those produced by Schottky groups of rank g; these are called Schottky structures. A Whittaker group of rank g is by definition a Kleinian group K containing, as an index two subgroup, a Schottky group Γ of rank g. In this case, K corresponds exactly to a conformal involution on the handlebody with Schottky structure given by Γ. In this paper we provide a structural description of Whittaker groups and, as a consequence of this, we obtain some facts concerning conformal involutions on handlebodies. For instance, we give a formula to count the type and the number of connected components of the set of fixed points of a conformal involution of a handlebody with a Schottky structure in terms of a group of automorphisms containing the conformal involution.