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


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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  1. Brown K.S.: Cohomology of Groups Graduate Texts in Mathematics, vol. 87. Springer, Berlin (1982)

    Google Scholar 

  2. Buser P.: Geometry and Spectra of Compact Riemann Surfaces Progress in Mathematics, vol. 106. Birkhäuser Boston Inc., Boston (1992)

    Google Scholar 

  3. Culler M., Shalen P.B.: Varieties of group representations and splittings of 3-manifolds. Ann. Math. (2) 117(1), 109–146 (1983). doi:10.2307/2006973

    Article  MathSciNet  MATH  Google Scholar 

  4. Eisenbud D., Hirsch U., Neumann W.: Transverse foliations of seifert bundles and self-homeomorphism of the circle. Comment. Math. Helv 56(4), 638–660 (1981). doi:10.1007/BF02566232

    Article  MathSciNet  MATH  Google Scholar 

  5. Gallo D., Kapovich M., Marden A.: The monodromy groups of Schwarzian equations on closed Riemann surfaces. Ann. Math. (2) 2(151), 625–704 (2000). doi:10.2307/121044

    Article  MathSciNet  Google Scholar 

  6. Goldman, W.M.: Discontinuous groups and the euler class. Ph.D. thesis, Berkeley (1980)

  7. Goldman W.M.: The symplectic nature of fundamental groups of surfaces. Adv. Math. 54(2), 200–225 (1984). doi:10.1016/0001-8708(84)90040-9

    Article  MATH  Google Scholar 

  8. Goldman W.M.: Topological components of spaces of representations. Invent. Math. 93(3), 557–607 (1988). doi:10.1007/BF01410200

    Article  MathSciNet  MATH  Google Scholar 

  9. Goldman W.M.: Ergodic theory on moduli spaces. Ann. Math. (3) 2(146), 475–507 (1997). doi:10.2307/2952454

    Article  Google Scholar 

  10. Goldman W.M.: The modular group action on real SL(2)-characters of a one-holed torus. Geom. Topol. 7, 443–486 (2003). doi:10.2140/gt.2003.7.443 (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hilton P.J., Stammbach U.: A Course in Homological Algebra, Graduate Texts in Mathematics, vol. 4, 2nd edn. Springer, New York (1997)

    Google Scholar 

  12. Hodgson, C.D.: Degeneration and regeneration of geometric structures on three-manifolds. Ph.D. thesis, Princeton University (1986)

  13. Huebschmann J.: Symplectic and Poisson structures of certain moduli spaces I. Duke Math. J. 80(3), 737–756 (1995). doi:10.1215/S0012-7094-95-08024-7

    Article  MathSciNet  MATH  Google Scholar 

  14. Leleu, X.: Géométries de courbure constante des 3-variétés et variétés de caractères de représentations dans \({sl_2(\mathbb{C})}\) . Ph.D. thesis, Université de Provence, Marseille (2000)

  15. Mathews, D.: From algebra to geometry: a hyperbolic odyssey; the construction of geometric cone-manifold structures with prescribed holonomy. Masters thesis, University of Melbourne (2005) Available at the author’s website,

  16. Mathews, D.: Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori (2010) Accepted for publication in Geometriae Dedicata, also available at

  17. Mathews, D.: The hyperbolic meaning of the Milnor–Wood inequality (2010) Submitted for publication, also available at

  18. Milnor J.: On the existence of a connection with curvature zero. Comment. Math. Helv. 32, 215–223 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  19. Milnor, J.W., Stasheff, J.D.: Characteristic classes. N. J. Annals of Mathematics Studies, Princeton University Press, Princeton, No. 76 (1974)

  20. Nielsen J.: Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. Acta Math. 50(1), 189–358 (1927). doi:10.1007/BF02421324

    Article  MathSciNet  MATH  Google Scholar 

  21. Pollard D.: A user’s guide to measure theoretic probability, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 8 . Cambridge University Press, Cambridge (2002)

    Google Scholar 

  22. Stillwell, J.: The Dehn-Nielsen theorem. In: Papers on Group Theory and Topology by Max Dehn. Springer, Berlin (1988)

  23. Tan S.P.: Branched CP 1-structures on surfaces with prescribed real holonomy. Math. Ann. 300(4), 649–667 (1994). doi:10.1007/BF01450507

    Article  MathSciNet  MATH  Google Scholar 

  24. Thurston, W.P.: The geometry and topology of 3-manifolds. Mimeographed notes (1979)

  25. Thurston W.P.: Three-dimensional geometry and topology, vol. 1. In: Levy, S. (eds) Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton (1997)

    Google Scholar 

  26. Troyanov M.: Prescribing curvature on compact surfaces with conical singularities. Trans. Am. Math. Soc. 324(2), 793–821 (1991). doi:10.2307/2001742

    Article  MathSciNet  MATH  Google Scholar 

  27. Wood J.W.: Bundles with totally disconnected structure group. Comment. Math. Helv. 46, 257–273 (1971)

    Article  MathSciNet  MATH  Google Scholar 

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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).

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  • Hyperbolic
  • Cone-manifold
  • Holonomy

Mathematics Subject Classification (2000)

  • 57M50