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arXiv · 2105.14948

From hyperbolic Dehn filling to surgeries in representation varieties

Georgios Kydonakis
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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 $\Sigma$ and a semisimple Lie group $G$. Explicit examples are produced in the case of $\Theta$-positive representations lying in the smooth connected components of the $\text{SO} \left(p,p+1 \right)$-representation variety.

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Georgios Kydonakis. 2021-05-31. From hyperbolic Dehn filling to surgeries in representation varieties. https://arxiv.org/abs/2105.14948

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