#### Volume 22, issue 5 (2018)

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 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 $\Gamma ↷X$ is a convex-cocompact action of a discrete group on a noncompact rank-one symmetric space $X\phantom{\rule{0.3em}{0ex}}$, there is a natural lower bound for the Hausdorff dimension of the limit set $\Lambda \left(\Gamma \right)\subset \partial X\phantom{\rule{0.3em}{0ex}}$, given by the Ahlfors regular conformal dimension of $\partial \Gamma \phantom{\rule{0.3em}{0ex}}$. 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.

##### Keywords
convex-cocompact action, rank-one symmetric space, Carnot group
##### Mathematical Subject Classification 2010
Primary: 53C24, 53C35
Secondary: 53C17, 53C23