From hyperbolic Dehn filling to surgeries in representation varieties

TL;DR

This paper explores the connections between hyperbolic Dehn filling, bending procedures, and gluing techniques in representation varieties, demonstrating how these methods can construct models in the context of Higgs bundles and specific Lie group representations.

Contribution

It establishes a logical link between hyperbolic geometric methods and Higgs bundle gluing techniques, providing explicit constructions in representation varieties for certain Lie groups.

Findings

Constructed models in representation varieties for surface groups and Lie groups.

Linked hyperbolic Dehn filling and Higgs bundle gluing methods.

Produced explicit examples of $ ext{SO}(p,p+1)$-positive representations.

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.