Volume 22, issue 5 (2018)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 22
Issue 5, 2511–3144
Issue 4, 1893–2510
Issue 3, 1267–1891
Issue 2, 645–1266
Issue 1, 1–644

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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.

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