Please use this identifier to cite or link to this item: http://scholarbank.nus.edu.sg/handle/10635/14470
Title: Hyperbolic cone-surfaces, generalized Markoff maps, Schottky groups and McShane's identity
Authors: ZHANG YING
Keywords: simple geodesic, McShane's identity, hyperbolic cone-surface, generalized Markoff map, Schottky group, torus bundle over ther circle
Issue Date: 25-Jan-2005
Source: ZHANG YING (2005-01-25). Hyperbolic cone-surfaces, generalized Markoff maps, Schottky groups and McShane's identity. ScholarBank@NUS Repository.
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.
URI: http://scholarbank.nus.edu.sg/handle/10635/14470
Appears in Collections:Ph.D Theses (Open)

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