论文标题
具有紧凑的开放亚组，相对双曲和连贯性的拓扑组
Topological groups with a compact open subgroup, Relative hyperbolicity and Coherence
论文作者
论文摘要
本文研究的主要对象是对$（g，\ nathcal {h}）$，其中$ g $是一个拓扑组，带有紧凑的开放子组，而$ \ Mathcal {h} $是开放子组的有限集合。我们开发了几何技术来研究$ g $的概念，并相对于$ \ Mathcal H $紧凑和紧凑。这包括$ g $在复合物上的离散动作的拓扑特征，当$ g $相对于$ \ MATHCAL H $紧密生成时，与Pairs $ $（G，G，\ Mathcal H）$相关的某些图形的准偶然不变性以及已知结果的扩展。例如，将OSIN的结果概括为离散组，我们表明，在$ G $相对于$ \ Mathcal H $的情况下，$ G $是紧凑的： $ \ bullet $如果$ g $是紧凑的，则\ Mathcal H $中的每个子组$ h \都是紧凑的； $ \ bullet $如果每个子组$ h \ in \ mathcal h $紧凑，则$ g $被紧凑。 本文还介绍了一种基于Bowditch的工作，使用双曲线细图上的离散操作，对Pairs $（G，\ Mathcal H）的相对双曲线进行了一种方法。例如，我们证明，如果$ g $相对于$ \ Mathcal H $是双曲线，则$ G $相对于$ \ Mathcal H $紧凑。作为本文结果的应用，我们证明了具有紧凑的开放子组的相干拓扑组的组合结果，并将麦卡蒙德的周边方法扩展到该一般框架。
The main objects of study in this article are pairs $(G, \mathcal{H})$ where $G$ is a topological group with a compact open subgroup, and $\mathcal{H}$ is a finite collection of open subgroups. We develop geometric techniques to study the notions of $G$ being compactly generated and compactly presented relative to $\mathcal H$. This includes topological characterizations in terms of discrete actions of $G$ on complexes, quasi-isometry invariance of certain graphs associated to the pairs $(G,\mathcal H)$ when $G$ is compactly generated relative to $\mathcal H$, and extensions of known results for the discrete case. For example, generalizing results of Osin for discrete groups, we show that in the case that $G$ is compactly presented relative to $\mathcal H$: $\bullet$ if $G$ is compactly generated, then each subgroup $H\in \mathcal H$ is compactly generated; $\bullet$ if each subgroup $H\in \mathcal H$ is compactly presented, then $G$ is compactly presented. The article also introduces an approach to relative hyperbolicity for pairs $(G, \mathcal H)$ based on Bowditch's work using discrete actions on hyperbolic fine graphs. For example, we prove that if $G$ is hyperbolic relative to $\mathcal H$ then $G$ is compactly presented relative to $\mathcal H$. As applications of the results of the article we prove combination results for coherent topological groups with a compact open subgroup, and extend McCammond-Wise perimeter method to this general framework.
