管状群的可公度性与拟等距分类：单顶点单环情形
Commensurability and quasi-isometry classification for one vertex one loop tubular groups
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中文总结 AI 辅助
本文对单顶点单环管状群按边映射交数分类：非零交数时全体可公度且拟等距；零交数时有两个拟等距类和无穷可公度类，两类间不拟等距。
中文摘要 AI 辅助
管状群具有一个图分解，其顶点群为$\mathbb{Z}^2$，边群为$\mathbb{Z}$。本文给出了单顶点单环管状群$G_{(k,\ell),(m,n)} = \langle a,b,t: [a,b]=1, t a^m b^n t^{-1} = a^k b^\ell \rangle$在可公度性和拟等距意义下的分类。我们证明了当边映射的像具有非零“交数”时，所有管状群彼此可公度，因此彼此拟等距。当交数为零时，存在两个拟等距类和无穷多个可公度类。非零交数与零交数的管状群之间不拟等距。
英文摘要
A tubular group has a graph of groups decomposition with $\mathbb{Z}^2$ vertex groups and $\mathbb{Z}$ edge groups. This paper gives a classification of one vertex one loop tubular groups $G_{(k,\ell),(m,n)} = \langle a,b,t: [a,b]=1, t a^m b^n t^{-1} = a^k b^\ell \rangle$ up to commensurability and quasi-isometry. We show that tubular groups where the images of the edge maps have nonzero ``intersection number" are all commensurable and so quasi-isometric. When the intersection number is zero, there are two quasi-isometry classes and infinitely many commensurability classes. The nonzero and zero intersection number tubular groups are not quasi-isometric.
