Geodesically tracking quasi‐geodesic paths for Coxeter groups
Geodesically tracking quasi‐geodesic paths for Coxeter groups
复制标题
测地线跟踪 Coxeter 群的准测地线路径
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
S. Tschantz
中科院分区：
文献类型：
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作者：
M. Mihalik;S. Tschantz
如果Λ是Gromov双曲群的Cayley图，则Λ中的拟测地线被测地线跟踪是一个基本事实。设（W，S）是一个非线性生成的Coxeter系统，Λ是（W，S）的Cayley图.对于一般Coxeter群，不是所有Λ中的拟测地线都被测地线跟踪。本文对由测地线跟踪的Λ-拟测地线射线进行了分类。作为推论，我们证明，如果W几何地作用在CAT（0）空间X上，则X中的CAT（0）测地线被凯莱图测地线跟踪（将凯莱图等变地放置在X中），并且对于任何A S，特殊子群<A>在X中是拟凸的。我们还证明了，如果g是（W，S）的无限阶元，则子群g ∈ G ∈ G被Λ中的Cayley测地线跟踪（类似于字双曲群的相应结果）.
If Λ is the Cayley graph of a Gromov hyperbolic group, then it is a fundamental fact that quasi‐geodesics in Λ are tracked by geodesics. Let (W, S) be a finitely generated Coxeter system and Λ be the Cayley graph of (W, S). For general Coxeter groups, not all quasi‐geodesic rays in Λ are tracked by geodesics. In this paper, we classify the Λ‐quasi‐geodesic rays that are tracked by geodesics. As corollaries we show that if W acts geometrically on a CAT(0) space X, then CAT(0) geodesics in X are tracked by Cayley graph geodesics (taking the Cayley graph as equivariantly placed in X) and for any A ⊂ S, the special subgroup 〈A〉 is quasi‐convex in X. We also show that if g is an element of infinite order for (W, S), then the subgroup 〈g〉 is tracked by a Cayley geodesic in Λ (in analogy with the corresponding result for word hyperbolic groups).