  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.
