全不连通局部紧群的图之间的拟等距
Quasi-isometries between graphs of totally disconnected locally compact groups
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- JLU Giessen（吉森大学）
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中文总结 AI 辅助
该研究将Papasoglu-Whyte的拟等距结果推广到全不连通局部紧群，证明此类群拟等距的充要条件，并构造了不可数个两两非拟等距的相关简单群，强化了已有结论。
中文摘要 AI 辅助
设$G$和$H$是紧生成的全不连通局部紧（tdlc）群，它们分别分解为tdlc群的有限图$(\boldsymbol{\textit{G}},\boldsymbol{\textit{A}})$和$(\boldsymbol{\textit{H}},\boldsymbol{\textit{B}})$，其中所有边群都是紧的，所有顶点群至多有一个端。我们将Papasoglu-Whyte的结果推广到tdlc群，证明$G$和$H$拟等距当且仅当它们具有相同的端数，且$(\boldsymbol{\textit{G}},\boldsymbol{\textit{A}})$的每个一端顶点群都与$(\boldsymbol{\textit{H}},\boldsymbol{\textit{B}})$的一个一端顶点群拟等距，反之亦然。作为应用，我们构造了不可数个两两非拟等距的紧生成非离散简单tdlc群，强化了Smith的结果。
英文摘要
Let $G$ and $H$ be compactly generated totally disconnected locally compact (tdlc) groups that decompose as finite graphs of tdlc groups $(\mathcal{G}, \mathcal{A})$ and $(\mathcal{H}, \mathcal{B})$ such that all edge groups are compact and all vertex groups have at most one end. We generalize a result of Papasoglu--Whyte to tdlc groups and show that $G$ and $H$ are quasi-isometric if and only if they have the same number of ends and every one-ended vertex group of $(\mathcal{G}, \mathcal{A})$ is quasi-isometric to a one-ended vertex group of $(\mathcal{H}, \mathcal{B})$, and vice versa. As an application, we construct uncountably many pairwise non-quasi-isometric compactly generated non-discrete simple tdlc groups, strengthening a result by Smith.
