Rigidity of warped cones and coarse geometry of expanders
Rigidity of warped cones and coarse geometry of expanders
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DOI：
10.1016/j.aim.2019.02.015
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发表时间：
2017-10
影响因子：
1.7
通讯作者：
D. Fisher;Thang Nguyen;Wouter van Limbeek
中科院分区：
文献类型：
--
作者：
D. Fisher;Thang Nguyen;Wouter van Limbeek
本文研究了自由极小等距群作用上的翘曲锥的几何性质以及相关的扩张图的构造。我们证明了一个刚性定理的粗糙几何的这种扭曲锥：即，如果一个组没有阿贝尔因子，那么两个这样的扭曲锥是拟等距的当且仅当行动是有限覆盖的共轭行动。因此，我们产生了连续的非拟等距扩张族和超扩张族。证明依赖于使用粗糙拓扑的扭曲锥，如计算其粗糙的基本群。
We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian factors, then two such warped cones are quasi-isometric if and only if the actions are finite covers of conjugate actions. As a consequence, we produce continuous families of non-quasi-isometric expanders and superexpanders. The proof relies on the use of coarse topology for warped cones, such as a computation of their coarse fundamental groups.