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Allen already said almost everything that there is to say...

I'll just add one more that he did not mention expicitelyexplicitly: $PSL(2,\mathbb R)\cong PSU(1,1) \cong SO(2,1)_+$.

This group is the group of Moebius transformations of the circle.
It's is also the group of conformal transformations of the disc, and also the isometry group of the hyperbolic plane. The above three names for this Lie group are related to three models of the hyperbolic plane: The upper half plane model, the Poincare disk model, and the upper sheet of hyperboloid model.
show/hide this revision's text 2 added 2 characters in body

Allen already said almost everything that there is to say...

I'll just add one more that he did not mention expicitely: $SL(2,\mathbb PSL(2,\mathbb R)\cong SU(1,1PSU(1,1) \cong SO(2,1)_+$.

This group is the group of Moebius transformations of the circle.
It's is also the group of conformal transformations of the disc, and also the isometry group of the hyperbolic plane. The above three names for this Lie group are related to three models of the hyperbolic plane: The upper half plane model, the Poincare disk model, and the upper sheet of hyperboloid model.
show/hide this revision's text 1

Allen already said almost everything that there is to say...

I'll just add one more that he did not mention expicitely: $SL(2,\mathbb R)\cong SU(1,1) \cong SO(2,1)_+$.

This group is the group of Moebius transformations of the circle.
It's is also the group of conformal transformations of the disc, and also the isometry group of the hyperbolic plane. The above three names for this Lie group are related to three models of the hyperbolic plane: The upper half plane model, the Poincare disk model, and the upper sheet of hyperboloid model.