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Hyperbolic cone-manifold structures with prescribed holonomy II: higher genus

Geometriae Dedicata volume 160, pages 15–45 (2012)Cite this article

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 build upon previous work with punctured tori to prove results for higher genus surfaces. Our techniques construct fundamental domains for hyperbolic cone-manifold structures, from the geometry of a representation. Central to these techniques are the Euler class of a representation, the group \({\widetilde{PSL_{2}\mathbb{R}}}\) , the twist of hyperbolic isometries, and character varieties. We consider the action of the outer automorphism and related groups on the character variety, which is measure-preserving with respect to a natural measure derived from its symplectic structure, and ergodic in certain regions. Under various hypotheses, we almost surely or surely obtain a hyperbolic cone-manifold structure with prescribed holonomy.

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Authors and Affiliations

  1. Department of Mathematics, Boston College, Chestnut Hill, MA, 02467, USA

    Daniel V. Mathews

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  1. Daniel V. Mathews
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Correspondence to Daniel V. Mathews.

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Mathews, D.V. Hyperbolic cone-manifold structures with prescribed holonomy II: higher genus. Geom Dedicata 160, 15–45 (2012). https://doi.org/10.1007/s10711-011-9668-y

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Keywords

  • Hyperbolic
  • Cone-manifold
  • Holonomy

Mathematics Subject Classification (2000)

  • 57M50