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From Hyperbolic Dehn Filling to Surgeries in Representation Varieties

In: In the Tradition of Thurston II

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  • Georgios Kydonakis

    (Alexander von Humboldt-Stiftung
    Ruprecht-Karls-Universität Heidelberg)

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 Hom π 1 Σ , G $$\text{Hom} \left ( \pi _{1} \left ( \Sigma \right ), G \right ) $$ 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 SO p , p + 1 $$\text{SO} \left (p,p+1 \right )$$ representation variety.

Suggested Citation

  • Georgios Kydonakis, 2022. "From Hyperbolic Dehn Filling to Surgeries in Representation Varieties," Springer Books, in: Ken’ichi Ohshika & Athanase Papadopoulos (ed.), In the Tradition of Thurston II, chapter 0, pages 201-260, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-97560-9_6
    DOI: 10.1007/978-3-030-97560-9_6
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