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

Abstract

When Γ X is a convex-cocompact action of a discrete group on a non-compact rank-one symmetric space X, there is a natural lower bound for the Hausdorff dimension of the limit set Λ(Γ) ⊂ ∂X, given by the Ahlfors regular conformal dimension of ∂Γ. We show that equality is achieved precisely when Γ stabilizes an isometric copy of some non-compact rank-one symmetric space in X on which it acts with compact quotient. This generalizes a theorem of Bonk-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 G. We show that almost all tangents are isometric to a Carnot subgroup of 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.

Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings

“…In other words, to admit a bi-Lipschitz embedding into a Euclidean space, a PI space must itself be infinitesimally Euclidean. (Extensions of this are known, see [16,Theorem 14.2] and more recent results in [17,18,19,21,23,53]. )…”