Rigidity for convex-cocompact actions on rank-one symmetric spaces
Rigidity for convex-cocompact actions on rank-one symmetric spaces
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DOI：
10.2140/gt.2018.22.2757
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发表时间：
2016-09
影响因子：
2
通讯作者：
Guy C. David;K. Kinneberg
中科院分区：
文献类型：
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作者：
Guy C. David;K. Kinneberg
当离散群在非紧的一阶对称空间上存在凸余紧作用时，极限集的Hausdorff维数存在一个自然的下界，这个下界由群边界的Ahlfors正则保形维数给出.我们证明，当群稳定其作用于紧商的某个非紧的秩一对称空间的等距副本时，等价性是精确实现的。这推广了Bonk-Kleiner的一个定理，他在实双曲空间中证明了这个定理。为了证明我们的主要定理，我们研究了嵌入到卡诺群中的Lipschitz可微空间的切线。我们证明了几乎所有的切线都等距于卡诺子群，至少当它们是可直连通的时候是这样。这推广了Cheeger的一个定理，他证明了嵌入在欧氏空间中的PI空间的一个定理。
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.