Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity
Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity
复制标题
DOI：
10.1007/s00208-008-0317-1
复制
发表时间：
2005-12
影响因子：
1.4
通讯作者：
Jason A. Behrstock;Cornelia Drutu;L. Mosher
中科院分区：
文献类型：
--
作者：
Jason A. Behrstock;Cornelia Drutu;L. Mosher
我们研究了非相对双曲群的几何。推广了Schwartz的一个结果，非相对双曲空间在相对双曲空间中的任何拟等距像都包含在单个外围子群的有界邻域中。这意味着一个群是相对双曲的，而其周边子群是非相对双曲的，则它是一个拟等距不变量。作为一个应用，阿廷群是相对双曲的当且仅当可自由分解。我们还引入了一个新的度量空间的拟等距不变量，称为度量厚，这是足够的度量空间是非双曲相对于任何非平凡的子集的集合。厚群生成的群包括：大多数曲面的映射类群;大多数自由群的外自同构群;某些Artin群;以及其他。高阶半单李群中的非均匀格是厚的，因此是非相对双曲的，与提供相对双曲群的激励性例子的秩1相反。映射类群是第一个例子，非相对双曲群的截点在任何渐近锥，解决了几个问题的Drutu和萨皮尔的结构相对双曲群。在群论之外，对于足够大的复杂度的曲面，Teichmüller空间相对于Weil-Peterson度量是厚的，与Brock-Farb的双曲性导致低复杂度相反。
We study the geometry of non-relatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with non-relatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces calledmetrically thick, which is sufficient for a metric space to be non-hyperbolic relative to any non-trivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Non-uniform lattices in higher rank semisimple Lie groups are thick and hence non-relatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of non-relatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmüller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil–Peterson metric, in contrast with Brock–Farb’s hyperbolicity result in low complexity.