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Dominating surface-group representations into \({\textrm{PSL}_2 (\mathbb {C})}\) in the relative representation variety

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Let \(\rho \) be a representation of the fundamental group of a punctured surface into \({\textrm{PSL}_2 (\mathbb {C})}\) that is not Fuchsian. We prove that there exists a Fuchsian representation that strictly dominates \(\rho \) in the simple length spectrum, and preserves the boundary lengths. This extends a result of Gueritaud-Kassel-Wolff to the case of \({\textrm{PSL}_2 (\mathbb {C})}\)-representations. Our proof involves straightening the pleated plane in \({\mathbb {H}}^3\) determined by the Fock-Goncharov coordinates of a framed representation, and applying strip-deformations.

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Acknowledgements

This paper was conceived during SG’s visit to Fudan University in June 2019; he is grateful for their hospitality and support. SG also thanks the SERB, DST (Grant No. MT/2017/000706), and the UGC Center for Advanced Studies Grant for their support. W. Su is partially supported by NSFC No: 11671092, 11631010, 11911530228. We are grateful to Nathaniel Sagman for his comments on an earlier version of this paper, and for pointing out how to handle the degenerate and co-axial case in §3.2. We are also grateful for the comments of anonymous referees on a previous versions of this paper.

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  1. Department of Mathematics, Indian Institute of Science, Bangalore, India

    Subhojoy Gupta

  2. School of Mathematics, Sun Yat-sen University, Guangzhou, China

    Weixu Su

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  1. Subhojoy Gupta
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Correspondence to Subhojoy Gupta.

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Gupta, S., Su, W. Dominating surface-group representations into \({\textrm{PSL}_2 (\mathbb {C})}\) in the relative representation variety. manuscripta math. 172, 1169–1186 (2023). https://doi.org/10.1007/s00229-022-01443-6

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  • DOI: https://doi.org/10.1007/s00229-022-01443-6

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