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 $\mathcal{D}_3$.