Algebraic Structure
The center of SL(2,R) is the two-element group {±1}, and the quotient PSL(2,R) is simple.
Discrete subgroups of PSL(2,R) are called Fuchsian groups. These are the hyperbolic analogue of the Euclidean wallpaper groups and Frieze groups. The most famous of these is the modular group PSL(2,Z), which acts on a tessellation of the hyperbolic plane by ideal triangles.
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.
Read more about this topic: SL2(R)
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