Parabolic Elements
A parabolic element has only a single eigenvalue, which is either 1 or -1. Such an element acts as a shear mapping on the Euclidean plane, and the corresponding element of PSL(2,R) acts as a limit rotation of the hyperbolic plane and as a null rotation of Minkowski space.
Parabolic elements of the modular group act as Dehn twists of the torus.
Parabolic elements are conjugate into the 2 component group of standard shears × ±I: . In fact, they are all conjugate (in SL(2)) to one of the four matrices, (in GL(2) or SL±(2), the ± can be omitted, but in SL(2) it cannot).
Read more about this topic: SL2(R), Classification of Elements
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