caa a22 4500 445884630 CHVBK 20180317145555.0 cr unu---uuuuu 170323e20110601xx s 000 0 eng 10.1007/s10711-010-9547-y doi (NATIONALLICENCE)springer-10.1007/s10711-010-9547-y Mathews Daniel Département de mathématiques, Université de Nantes, 2, rue de la Houssinire, BP 92208, 44322, Nantes Cedex 3, France aut Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori [Elektronische Daten] [Daniel Mathews] We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of 2π, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we prove results for the punctured torus; in the sequel, for higher genus surfaces. We show that a representation of the fundamental group of a punctured torus is a holonomy representation of a hyperbolic cone-manifold structure with no interior cone points and a single corner point if and only if it is not virtually abelian. We construct a pentagonal fundamental domain for hyperbolic structures, from the geometry of a representation. Our techniques involve the universal covering group $${\widetilde{{\it PSL}_2{\mathbb R}}}$$ of the group of orientation-preserving isometries of $${{\mathbb H}^2}$$ and Markoff moves arising from the action of the mapping class group on the character variety. Springer Science+Business Media B.V., 2011 Hyperbolic nationallicence Cone-manifold nationallicence Holonomy nationallicence Geometriae Dedicata Springer Netherlands 152/1(2011-06-01), 85-128 0046-5755 152:1<85 2011 152 10711 https://doi.org/10.1007/s10711-010-9547-y text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10711-010-9547-y text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Mathews Daniel Département de mathématiques, Université de Nantes, 2, rue de la Houssinire, BP 92208, 44322, Nantes Cedex 3, France aut NATIONALLICENCE 773 0- Geometriae Dedicata Springer Netherlands 152/1(2011-06-01), 85-128 0046-5755 152:1<85 2011 152 10711 Metadata rights reserved Springer special CC-BY-NC licence nationallicence SWISSBIB 445884630 BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer