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Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori

Abstract

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.

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Correspondence to Daniel V. Mathews.

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Mathews, D.V. Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori. Geom Dedicata 152, 85–128 (2011). https://doi.org/10.1007/s10711-010-9547-y

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Keywords

  • Hyperbolic
  • Cone-manifold
  • Holonomy

Mathematics Subject Classification (2000)

  • 57M50