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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.