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

Guy C David and Kyle Kinneberg

Geometry & Topology 22 (2018) 2757–2790

When Γ 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 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 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 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.

convex-cocompact action, rank-one symmetric space, Carnot group
Mathematical Subject Classification 2010
Primary: 53C24, 53C35
Secondary: 53C17, 53C23
Received: 15 September 2016
Revised: 26 January 2018
Accepted: 25 February 2018
Published: 1 June 2018
Proposed: Bruce Kleiner
Seconded: Benson Farb, John Lott
Guy C David
Department of Mathematical Sciences
Ball State University
Muncie, IN
United States
Courant Institute of Mathematical Sciences
New York University
New York, NY
Kyle Kinneberg
National Security Agency
Fort Meade, MD
United States
Department of Mathematics
Rice University
Houston, TX
United States