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From Hyperbolic Dehn Filling to Surgeries in Representation Varieties

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In the Tradition of Thurston II

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Abstract

Hyperbolic Dehn surgery and the bending procedure provide two ways which can be used to describe hyperbolic deformations of a complete hyperbolic structure on a 3-manifold. Moreover, one can obtain examples of non-Haken manifolds without the use of Thurston’s Uniformization Theorem. We review these gluing techniques and present a logical continuity between these ideas and gluing methods for Higgs bundles. We demonstrate how one can construct certain model objects in representation varieties \(\text{Hom} \left ( \pi _{1} \left ( \Sigma \right ), G \right ) \) for a topological surface Σ and a semisimple Lie group G. Explicit examples are produced in the case of Θ-positive representations lying in the smooth connected components of the \(\text{SO} \left (p,p+1 \right )\) representation variety.

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Acknowledgements

I would like to warmly thank the editors Ken’ichi Ohshika and Athanase Papadopoulos for their kind invitation to contribute to this collective volume. The work included in Sect. 6.7 was supported by the Labex IRMIA of the Université de Strasbourg and was undertaken in collaboration with Olivier Guichard. I acknowledge the Max-Planck-Institut für Mathematik in Bonn for its hospitality and support during the production of this chapter. I am also grateful to an anonymous referee as well as the editors for their careful reading of the manuscript and a number of useful suggestions.

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  1. Alexander von Humboldt-Stiftung, Bonn, Germany

    Georgios Kydonakis

  2. Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

    Georgios Kydonakis

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  1. Georgios Kydonakis
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  1. Department of Mathematics, Gakushuin University, Tokyo, Japan

    Ken’ichi Ohshika

  2. Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, Strasbourg, France

    Athanase Papadopoulos

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Kydonakis, G. (2022). From Hyperbolic Dehn Filling to Surgeries in Representation Varieties. In: Ohshika, K., Papadopoulos, A. (eds) In the Tradition of Thurston II. Springer, Cham. https://doi.org/10.1007/978-3-030-97560-9_6

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