Skip to main content

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


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.

This is a preview of subscription content, access via your institution.


  1. Bowditch, B.H.: A proof of McShane’s identity via Markoff triples. Bull. London Math. Soc. 28(1), 73–78 (1996). doi:10.1112/blms/28.1.73.

  2. Bowditch, B.H.: Markoff triples and quasi-Fuchsian groups. Proc. London Math. Soc. (3) 77(3), 697–736 (1998). doi:10.1112/S0024611598000604.

  3. Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)

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

  5. Cooper, D., Culler, M., Gillet, H., Long, D.D., Shalen, P.B.: Plane curves associated to character varieties of 3-manifolds. Invent. Math. 118(1), 47–84 (1994). doi:10.1007/BF01231526.

    Google Scholar 

  6. Cooper, D., Hodgson, C.D., Kerckhoff, S.P.: Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs, vol. 5. Mathematical Society of Japan, Tokyo (2000). With a postface by Sadayoshi Kojima

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

  8. Dunfield, N.M.: Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds. Invent. Math. 136(3), 623–657 (1999). doi:10.1007/s002220050321.

  9. 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.

  10. Francaviglia, S.: Hyperbolicity equations for cusped 3-manifolds and volume-rigidity of representations. Ph.D. thesis, Scuola Normale Superiore Pisa (2003)

  11. Fricke, R.: Über die theorie der automorphen modulgrupper. Nachr. Akad. Wiss. Göttingen, pp. 91–101 (1896)

  12. Fricke, R., Klein, F.: Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, Bibliotheca Mathematica Teubneriana, Bände 3, vol. 4. Johnson Reprint Corp., New York (1965)

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

  14. Gilman, J., Maskit, B.: An algorithm for 2-generator Fuchsian groups. Michigan Math. J. 38(1), 13–32 (1991). doi:10.1307/mmj/1029004258.

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

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

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

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

    Google Scholar 

  19. Hoste, J., Shanahan, P.D.: Trace fields of twist knots. J. Knot Theory Ramifications 10(4), 625–639 (2001). doi:10.1142/S0218216501001049.

    Google Scholar 

  20. 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)

  21. Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1977 edition

  22. Maclachlan, C., Reid, A.W.: The Arithmetic of Hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol. 219. Springer, New York (2003)

  23. Magnus, W.: Rings of Fricke characters and automorphism groups of free groups. Math. Z. 170(1), 91–103 (1980). doi:10.1007/BF01214715.

  24. Magnus, W., Karrass, A., Solitar, D.: Combinatorial group theory, second edn. Dover Publications Inc., Mineola, NY (2004). Presentations of groups in terms of generators and relations

  25. Matelski, J.P.: The classification of discrete 2-generator subgroups of PSL(2, R). Israel J. Math. 42(4), 309–317 (1982). doi:10.1007/BF02761412.

  26. Mathews, D.: Mahler’s unfinished symphony: Etudes in knots, algebra and geometry (2003). Honours project, University of Melbourne, 2003. Available at the author’s website,

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

  28. Mathews, D.: Hyperbolic cone-manifold structures with prescribed holonomy II: higher genus (2010).

  29. Mathews, D.: The hyperbolic meaning of the Milnor–Wood inequality (2010).

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  32. Nielsen, J.: Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden. Math. Ann. 78(1), 385–397 (1964). doi:10.1007/BF01457113.

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

  34. Tan, S.P.: Branched CP1-structures on surfaces with prescribed real holonomy. Math. Ann. 300(4), 649–667 (1994). doi:10.1007/BF01450507.

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

  36. Thurston, W.P.: Three-Dimensional Geometry and Topology. Vol. 1, Princeton Mathematical Series, vol. 35. Princeton University Press, Princeton, NJ (1997). Edited by Silvio Levy

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Daniel V. Mathews.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mathews, D.V. Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori. Geom Dedicata 152, 85–128 (2011).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Hyperbolic
  • Cone-manifold
  • Holonomy

Mathematics Subject Classification (2000)

  • 57M50