$SL_2(\boldsymbol{\text{Z}}[1/p])$的交换子中心化子与分类空间
Centralizers and classifying spaces for commutativity of $SL_2(\mathbb{Z}[1/p])$
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中文总结 AI 辅助
该研究计算了$SL_2(\boldsymbol{\text{Z}}[1/p])$在两种拓扑下的交换分类空间同伦型，利用几何群论明确其极大阿贝尔子群为非中心元素的中心化子，推进了群的拓扑分类研究。
中文摘要 AI 辅助
我们计算了由Adem、F. Cohen和Torres-Giese引入的、针对群$SL_2(\boldsymbol{\text{Z}}[1/p])$的交换分类空间的同伦型，分别将该群视为离散拓扑群和具有$SL_2(\boldsymbol{\text{R}})$的子空间拓扑。这需要整理$SL_2(\boldsymbol{\text{Z}}[1/p])$的极大阿贝尔子群的详细信息，这些子群恰好是该群非中心元素的中心化子，为此我们采用了几何群论的方法。
英文摘要
We compute the homotopy type of the classifying space for commutativity introduced by Adem, F. Cohen and Torres-Giese for the group $\SL_2(\Z[1/p])$, both as a discrete topological group and with the subspace topology from $\SL_2(\R)$. Doing so requires compiling detailed information on the maximal abelian subgroups of $\SL_2(\Z[1/p])$, which turn out to be precisely the centralizers of the non-central elements of the group, for which we employ methods from geometric group theory.
