大规模群体的结束
Ends of large scale groups
摘要：
本文的目的是将有限生成群的端理论与局部紧的、可度量化的、连通的拓扑群的端理论统一起来。在这两种理论中，人们都证明，如果端点的数目是有限的，那么它必须至多是$2$。在这两种理论中，两端的群都被刻画为具有一个无限循环子群，该子群要么具有有限指数，要么其陪集空间是紧的。我们的推广相当于定义任何粗糙空间的端点空间，然后将其应用到大规模群，一类群推广了有限生成群和局部紧的，可度量化的和连通的拓扑群。此外，我们证明了一个版本的Svarc-Milnor引理的大规模群体，我们证明了粗双曲大规模群体有有限的渐近维数，只要他们有界几何。
英文摘要：
The aim of this paper is to unify the theory of ends of finitely generated groups with that of ends of locally compact, metrizable and connected topological groups. In both theories one proves that, if the number of ends is finite, then it must be at most $2$. In both theories groups of two ends are characterized as having an infinite cyclic subgroup of either finite index or such that its coset space is compact. Our generalization amounts to defining the space of ends of any coarse space and then applying it to large scale groups, a class of groups generalizing both finitely generated groups and locally compact, metrizable and connected topological groups. Additionally, we prove a version of Svarc-Milnor Lemma for large scale groups and we prove that coarsely hyperbolic large scale groups have finite asymptotic dimension provided they have bounded geometry.
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