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

David, Guy¹ ; Kinneberg, Kyle²

¹ Department of Mathematical Sciences, Ball State University, Muncie, IN, United States, Courant Institute of Mathematical Sciences, New York University, New York, NY
² National Security Agency, Fort Meade, MD, United States, Department of Mathematics, Rice University, Houston, TX, United States

Geometry & topology, Tome 22 (2018) no. 5, pp. 2757-2790.

Résumé

When $\Gamma \curvearrowright X$ 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 $\Lambda \left(\Gamma \right)\subset \partial X$, given by the Ahlfors regular conformal dimension of $\partial \Gamma$. We show that equality is achieved precisely when $\Gamma$ 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 $\mathbb{G}$. We show that almost all tangents are isometric to a Carnot subgroup of $\mathbb{G}$, 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.

Détail

DOI : 10.2140/gt.2018.22.2757

Classification : 53C24, 53C35, 53C17, 53C23
Keywords: convex-cocompact action, rank-one symmetric space, Carnot group

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