Combinations and chromatography of paths of Kleinian groups

Abstract

We study continuous paths in the Chabauty topology on the set ${\mathcal{D}}_n$ of torsion free discrete subgroups of the isometry group of $n$-dimensional hyperbolic space. We prove a combination theorem for paths in ${\mathcal{D}}_n$, which allows us to construct an exotic path of discrete subgroups along which no two subgroups are isomorphic. We also introduce a technique we refer to as "chromatography" to prove a decomposition theorem that characterizes paths of convex cocompact groups in ${\mathcal{D}}_3$.

31 pages, 3 figures. Subsection 4.2 was split from v1 of arXiv:2410.15537 and heavily revised