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arXiv 2402.080742024-02math.KT

关于 $\textrm{SL}_2$ 与 $\textrm{PSL}_2$ 的低维同调群之间的联系

On the connections between the low dimensional homology groups of $\textrm{SL}_2$ and $\textrm{PSL}_2$

Behrooz Mirzaii, Elvis Torres Pérez

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AI总结：

该研究借助自然满射，探讨整环上特殊线性群 $\textrm{SL}_2(A)$ 与射影特殊线性群 $\textrm{PSL}_2(A)$ 的低维同调群关联，重点分析两类整环上的前三阶同调群联系。

摘要：

本文研究特殊线性群 $\textrm{SL}_2(A)$ 与射影特殊线性群 $\textrm{PSL}_2(A)$（其中 $A$ 为整环）的低维同调群，借助自然满射 $\textrm{SL}_2(A) \to \textrm{PSL}_2(A)$ 展开分析。我们具体研究了这些群在欧几里得整环 $\text{Z}[\frac{1}{m}]$（$m$ 为无平方因子整数）与局部整环上的第一、第二、第三同调群的联系。

英文摘要：

In this article we study the low dimensional homology groups of the special linear group $\textrm{SL}_2(A)$ and the projective special linear group $\textrm{PSL}_2(A)$, $A$ a domain, through the natural surjective map $\textrm{SL}_2(A) \to \textrm{PSL}_2(A)$. In particular, we study the connection of the first, the second and the third homology groups of these groups over euclidean domains $\mathbb{Z}[\frac{1}{m}]$, $m$ a square free integer, and local domains.

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