2010/06/27 by Daniel V. Mathews, Mathews, Daniel V.
Mathematics · #57M50 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1006.5223
openalex publication_date 2010/06/27 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We consider the relationship between hyperbolic cone-manifold structures on\nsurfaces, and algebraic representations of the fundamental group into a group\nof isometries. A hyperbolic cone-manifold structure on a surface, with all\ninterior cone angles being integer multiples of 2\π, determines a holonomy\nrepresentation of the fundamental group. We ask, conversely, when a\nrepresentation of the fundamental group is the holonomy of a hyperbolic\ncone-manifold structure. In this paper we prove results for the punctured\ntorus; in the sequel, for higher genus surfaces. We show that a representation\nof the fundamental group of a punctured torus is a holonomy representation of a\nhyperbolic cone-manifold structure with no interior cone points and a single\ncorner point if and only if it is not virtually abelian. We construct a\npentagonal fundamental domain for hyperbolic structures, from the geometry of a\nrepresentation. Our techniques involve the universal covering group of the\ngroup of orientation-preserving isometries of the hyperbolic plane, and Markoff\nmoves arising from the action of the mapping class group on the character\nvariety.\n