Rigidity for convex-cocompact actions on rank-one symmetric spaces

TL;DR

This paper establishes a rigidity theorem for convex-cocompact groups acting on rank-one symmetric spaces, linking the Hausdorff dimension of the limit set to the group's geometric stabilization properties.

Contribution

It generalizes Bonk-Kleiner's theorem from real hyperbolic space to all rank-one symmetric spaces, connecting boundary dimension equality with geometric stabilization.

Findings

Equality in the Hausdorff dimension bound characterizes stabilization of symmetric spaces.

Almost all tangents of Lipschitz differentiability spaces embedded in Carnot groups are isometric to Carnot subgroups.

Extension of Cheeger's theorem to spaces embedded in Carnot groups.

Abstract

When a discrete group admits a convex-cocompact action on a non-compact rank-one symmetric space, there is a natural lower bound for the Hausdorff dimension of the limit set, given by the Ahlfors regular conformal dimension of the boundary of the group. We show that equality is achieved precisely when the group stabilizes an isometric copy of some non-compact rank-one symmetric space on which it acts with compact quotient. This generalizes a theorem of Bonk-Kleiner, who proved it in the case of real hyperbolic space. To prove our main theorem, we study tangents of Lipschitz differentiability spaces that are embedded in a Carnot group. We show that almost all tangents are isometric to Carnot subgroups, at least when they are rectifiably connected. This extends a theorem of Cheeger, who proved it for PI spaces that are embedded in Euclidean space.