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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
Abstract

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.

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Keywords
convex-cocompact action, rank-one symmetric space, Carnot group
Mathematical Subject Classification 2010
Primary: 53C24, 53C35
Secondary: 53C17, 53C23
References
Publication
Received: 15 September 2016
Revised: 26 January 2018
Accepted: 25 February 2018
Published: 1 June 2018
Proposed: Bruce Kleiner
Seconded: Benson Farb, John Lott
Authors
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
https://sites.google.com/view/gcdavid
Kyle Kinneberg
National Security Agency
Fort Meade, MD
United States
Department of Mathematics
Rice University
Houston, TX
United States
https://sites.google.com/site/kekinneberg