From Hyperbolic Dehn Filling to Surgeries in Representation Varieties

Abstract

AbstractHyperbolic 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.KeywordsHyperbolic Dehn surgeryCharacter varietyHigher Teichmüller spaceHiggs bundleParabolic structureElliptic operatorAMS ClassificationPrimary: 53C07; Secondary: 14H6058D27