Special elliptic isometries, relative SU(2,1)-character varieties, and bendings

Classification of isometries of spaces of constant curvature and invariant subspaces

We study the varieties of invariant totally geodesic submanifolds of isometries of the spherical, Euclidean and hyperbolic spaces in each finite dimension. We show that the dimensions of the connected components of these varieties determine the orbit type (or the z-class) of the isometry. For this purpose, we introduce the Segre symbol of an isometry, a discrete invariant encoding the structure of its normal form, which parametrizes z-classes. We then provide a description of the isomorphism type of the varieties of invariant subspaces in terms of the Segre symbol.

From hyperbolic Dehn filling to surgeries in representation varieties

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( π_{1} \left( Σ\right), G \right) $ for a topological surface $Σ$ and a semisimple Lie…

具有复双曲结构的平凡丛的一个非平凡族

A non-trivial family of trivial bundles with complex hyperbolic structure

该研究在复双曲平面等距群PU(2,1)中分析特定对合空间的性质，构造了带锥点球面上欧拉数为零的圆盘轨形丛的4维弯曲连通复双曲结构族，为相关难题提供了更简洁的构造方案。