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Rigidity for
convex-cocompact actions on rank-one symmetric spaces
Guy C David and Kyle Kinneberg |
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Geometry & Topology 22 (2018) 2757–2790
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Abstract |
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When is a convex-cocompact action of a discrete group on a noncompact 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 . We show that equality is achieved precisely when stabilizes an isometric copy of some noncompact rank-one symmetric space in on which it acts with compact quotient. This generalizes a theorem of Bonk and Kleiner, who proved it in the case that is real hyperbolic. 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 a Carnot subgroup of , 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
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Mathematical Subject Classification 2010
Primary: 53C24, 53C35
Secondary: 53C17, 53C23
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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
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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 | |
