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

Abstract

In this article we study the low dimensional homology groups of the special linear group ${\text{SL}}_2(A)$ and the projective special linear group ${\text{PSL}}_2(A)$, $A$ a domain, through the natural surjective map ${\text{SL}}_2(A)\to {\text{PSL}}_2(A)$. 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.