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

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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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References

  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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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, http://www.danielmathews.info

  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, http://www.danielmathews.info

  28. Mathews, D.: Hyperbolic cone-manifold structures with prescribed holonomy II: higher genus (2010). http://arxiv.org/abs/1006.5384

  29. Mathews, D.: The hyperbolic meaning of the Milnor–Wood inequality (2010). http://arxiv.org/abs/1006.5403

  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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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. http://dx.doi.org/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 

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  1. Département de mathématiques, Université de Nantes, 2, rue de la Houssinire, BP 92208, 44322, Nantes Cedex 3, France

    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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