Dehn filling in semisimple Lie groups
Theodore Weisman
TL;DR
This work generalizes Thurston’s hyperbolic Dehn filling to arbitrary-rank semisimple Lie groups by developing an extended Dehn filling theory for extended geometrically finite (EGF) representations. It introduces a robust framework using relatively hyperbolic groups, Dehn filling spaces, and a relative quasi-geodesic automaton to control deformations, ensuring that deformations of EGF representations remain EGF and exhibit strong geometric convergence. The main theorem asserts that nearby deformations along extended Dehn filling spaces descend to Dehn-filled groups with guideposts such as boundary extensions and convergent limit sets; it also yields criteria for obtaining Anosov representations via filling and provides semicontinuity of limit sets. The results unify and extend several higher-rank Dehn filling phenomena, with concrete applications to relatively Anosov representations, rank-one cases, and convex projective geometry, offering new pathways to construct and study higher-rank geometric structures and their limit behavior.
Abstract
We generalize one part of Thurston's hyperbolic Dehn filling theorem to arbitrary-rank semisimple Lie groups by showing that certain deformations of extended geometrically finite subgroups of a semisimple Lie group are still extended geometrically finite. As a special case, our theorem gives a criterion which guarantees that a deformation of a relatively Anosov subgroup is (non-relatively) Anosov, and also ensures that limit sets vary continuously. Our result also applies to several higher-rank examples in convex projective geometry which are outside of the relatively Anosov setting.