 The circle group SO(2) is a maximal compact subgroup of SL(2,R), and the circle SO(2)/{±1} is a maximal compact subgroup of PSL(2,R).The Schur multiplier of the discrete group PSL(2,R) is much larger than Z, and the universal central extension is much larger than the universal covering group.However these large central extensions do not take the topology into account and are somewhat pathological.