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

Guy C. David; Kyle Kinneberg

doi:10.2140/gt.2018.22.2757

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arXiv · 1609.02975

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

Guy C. David·Kyle Kinneberg

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Abstract

When a discrete group admits a convex-cocompact action on a non-compact 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 the boundary of the group. We show that equality is achieved precisely when the group stabilizes an isometric copy of some non-compact rank-one symmetric space on which it acts with compact quotient. This generalizes a theorem of Bonk-Kleiner, who proved it in the case of real hyperbolic space. 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 Carnot subgroups, 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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BibTeX RIS

Guy C. David, Kyle Kinneberg. 2016-09-09. Rigidity for convex-cocompact actions on rank-one symmetric spaces. https://doi.org/10.2140/gt.2018.22.2757

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