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

Abstract

When $\mathrm{\Gamma }\curvearrowright X$ is a convex-cocompact action of a discrete group on a noncompact rank-one symmetric space $X$, there is a natural lower bound for the Hausdorff dimension of the limit set $\mathrm{\Lambda }\left(\mathrm{\Gamma }\right)\subset \partial X$, given by the Ahlfors regular conformal dimension of $\partial \mathrm{\Gamma }$. We show that equality is achieved precisely when $\mathrm{\Gamma }$ stabilizes an isometric copy of some noncompact rank-one symmetric space in $X$ on which it acts with compact quotient. This generalizes a theorem of Bonk and Kleiner, who proved it in the case that $X$ is real hyperbolic.

To prove our main theorem, we study tangents of Lipschitz differentiability spaces that are embedded in a Carnot group $\mathbb{G}$. We show that almost all tangents are isometric to a Carnot subgroup of $\mathbb{G}$, 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.