Coarse median spaces and groups
Coarse median spaces and groups
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DOI：
10.2140/pjm.2013.261.53
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发表时间：
2013-02
影响因子：
0.6
通讯作者：
B. Bowditch
中科院分区：
文献类型：
--
作者：
B. Bowditch
我们在度量空间上引入了粗中值的概念。这满足中值代数的公理，直到有界距离。测地线空间上这种中值的存在是准等距不变的，因此它适用于有限生成的群，通过它们的Cayley图。证明了这类空间的渐近锥是拓扑中值代数。我们定义了粗中位数的秩概念，并证明了它限定了空间中拟等距嵌入的欧几里得平面的维度。利用Behrstock和Minsky的质心结构，我们证明了映射类群具有这一性质，并恢复了Behrstock和Minsky和Hamenstadt的秩定理。我们探索了这类空间的各种其他性质，并发展了一些关于中值代数的背景材料。
We introduce the notion of a coarse median on a metric space. This satisfies the axioms of a median algebra up to bounded distance. The existence of such a median on a geodesic space is quasi-isometry invariant, and so it applies to finitely generated groups via their Cayley graphs. We show that asymptotic cones of such spaces are topological median algebras. We define a notion of rank for a coarse median and show that this bounds the dimension of a quasi-isometrically embedded euclidean plane in the space. Using the centroid construction of Behrstock and Minsky, we show that the mapping class group has this property, and recover the rank theorem of Behrstock and Minsky and of Hamenstadt. We explore various other properties of such spaces, and develop some of the background material regarding median algebras.