Hyperbolic cone-surfaces, generalized Markoff maps, Schottky groups and McShane's identity

Abstract

In this thesis we study hyperbolic cone-surfaces with cusps and/or geodesic boundary and obtain generalizations of McShane's identity concerning the lengths of certain simple closed geodesics on a cusped hyperbolic surface to identities for hyperbolic cone-surfaces. We obtain a unified identity in terms of the complex lengths of the geometric boundary components and give various applications. We further show that the identity extends to classical Schottky groups, giving some new identities for hyperbolic surfaces of fuchsian Schottky groups. We also generalize our identity for one-hole tori to an identity for representations of the once-punctured torus group into PSL(2,C) which satisfy certain conditions by studying generalized Markoff maps and indicate connections with the study of hyperbolic Dehn surgery on some incomplete hyperbolic 3-manifolds which are once-punctured torus bundles over the circle.