Relative rigidity, quasiconvexity and $C$–complexes
Relative rigidity, quasiconvexity and $C$–complexes
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DOI：
10.2140/agt.2008.8.1691
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发表时间：
2007-04
影响因子：
0.7
通讯作者：
Mahan Mj
中科院分区：
文献类型：
--
作者：
Mahan Mj
我们引入并研究了对(X，J)的相对刚性的概念，其中1)X是双曲度量空间，J是拟凸集的集合；2)X是相对双曲群，J是抛物线的集合；3)X是高阶对称空间，J是极大平坦的等变集合，相对刚性大致可以描述为将两个这样的J之间的一致真映射提升到相应的X之间的拟等距映射。一个相关的概念是C-复形，它是Tits复形在这方面的适应。我们证明了上述配对集合(X，J)的相对刚性。这推广了Schwarz关于双曲空间中测地线对称模式的一个结果。我们证明了一致真映射诱导相应的C-复形同构。给出了双曲群的子群拟凸性的几个刻画。
We introduce and study the notion of relative rigidity for pairs (X, J ) where 1) X is a hyperbolic metric space and J a collection of quasiconvex sets 2) X is a relatively hyperbolic group and J the collection of parabolics 3) X is a higher rank symmetric space and J an equivariant collection of maximal flats Relative rigidity can roughly be described as upgrading a uniformly proper map between two such J's to a quasi-isometry between the corresponding X's. A related notion is that of a C-complex which is the adaptation of a Tits complex to this context. We prove the relative rigidity of the collection of pairs (X, J ) as above. This generalises a result of Schwarz for symmetric patterns of geodesics in hyperbolic space. We show that a uniformly proper map induces an isomorphism of the corresponding C-complexes. We also give a couple of characterizations of quasiconvexity of subgroups of hyperbolic groups on the way.