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Geometrisation of purely hyperbolic representations in PSL2R

Gianluca Faraco
From the journal Advances in Geometry


Let S be a surface of genus g at least 2. A representation ρ:π1SPSL2R is said to be purely hyperbolic if its image consists only of hyperbolic elements along with the identity. We may wonder under which conditions such representations arise as the holonomy of a branched hyperbolic structure on S. In this work we characterise them completely, giving necessary and sufficient conditions.

MSC 2010: 57M50


I would like to thank my advisor Stefano Francaviglia for introducing me to this theory and for his constant encouragement. His advice and suggestions have been highly valuable. I also would like to thank Lorenzo Ruffoni for useful comments and suggestions about this work. Finally, I would like to thank the anonymous referee(s) for many useful comments and suggestions.

  1. Communicated by: J. Ratcliffe


[1] A. F. Beardon, The geometry of discrete groups Springer 1983. MR698777 Zbl 0528.30001Search in Google Scholar

[2] G. Calsamiglia, B. Deroin, S. Francaviglia, Branched projective structures with Fuchsian holonomy. Geom. Topol. 18 (2014), 379–446. MR3159165 Zbl 1286.30031Search in Google Scholar

[3] D. Cooper, C. D. Hodgson, S. P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds volume 5 of MSJ Memoirs Mathematical Society of Japan, Tokyo 2000. MR1778789 Zbl 0955.57014Search in Google Scholar

[4] A. L. Edmonds, Deformation of maps to branched coverings in dimension two. Ann. of Math. (2) 110 (1979), 113–125. MR541331 Zbl 0424.57002Search in Google Scholar

[5] G. Faraco, Geometrization of almost extremal representation in PSL2ℝ. Preprint 2018, arXiv:1802.00755Search in Google Scholar

[6] L. Funar, M. Wolff, Non-injective representations of a closed surface group into PSL(2,R). Math. Proc. Cambridge Philos. Soc. 142 (2007), 289–304. MR2314602 Zbl 1117.20034Search in Google Scholar

[7] D. Gallo, M. Kapovich, A. Marden, The monodromy groups of Schwarzian equations on closed Riemann surfaces. Ann. of Math. (2) 151 (2000), 625–704. MR1765706 Zbl 0977.30028Search in Google Scholar

[8] W. M. Goldman, Projective structures with Fuchsian holonomy. J. Differential Geom. 25 (1987), 297–326. MR882826 Zbl 0595.57012Search in Google Scholar

[9] W. M. Goldman, Topological components of spaces of representations. Invent. Math. 93 (1988), 557–607. MR952283 Zbl 0655.57019Search in Google Scholar

[10] S. Katok, Fuchsian groups University of Chicago Press, Chicago, IL 1992. MR1177168 Zbl 0753.30001Search in Google Scholar

[11] J. Marché, M. Wolff, The modular action on PSL2(R)-characters in genus 2. Duke Math. J. 165 (2016), 371–412. MR3457677 Zbl 1353.37056Search in Google Scholar

[12] D. V. Mathews, Hyperbolic cone-manifold structures with prescribed holonomy II: higher genus. Geom. Dedicata 160 (2012), 15–45. MR2970041 Zbl 1256.57014Search in Google Scholar

[13] M. Spivak, A comprehensive introduction to differential geometry. Vol. I–V. Third edition with corrections. Publish or Perish, Houston, TX 1999. Zbl 1213.53001Search in Google Scholar

[14] S. P. Tan, Branched CP1-structures on surfaces with prescribed real holonomy. Math. Ann. 300 (1994), 649–667. MR1314740 Zbl 0833.53054Search in Google Scholar

Received: 2018-10-02
Revised: 2019-02-27
Published Online: 2021-01-22
Published in Print: 2021-01-27

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