# Supergroups of special linear group:SL(2,R)

## Contents

The fundamental group of $SL(2,\R)$, viewed as a topological group, is the group of integers. This is therefore also the second cohomology group. The upshot is that the Schur covers of this group are the same as the topological covers, and these correspond to all the possible quotient groups of the fundamental group.
The universal covering group is a group with central subgroup $\mathbb{Z}$ and quotient SL(2,R) (see universal covering group of SL(2,R)). In addition, for every positive integer $n$, there is a $n$-fold cover with central subgroup cyclic of order $n$ having quotient $SL(2,\R)$ (the actual center is cyclic of order $2n$ and the inner automorphism group is PSL(2,R)).
Of particular interest is the case $n = 2$. The 2-fold cover of $SL(2,\R)$ is metaplectic group:Mp(2,R). This has center isomorphic to cyclic group:Z4 and inner autmoorphism group isomorphic to PSL(2,R).