Sorry if this is too elementary for this forum:
We have two models of 2 dimensional hyperbolic space:
The upper sheet of a hyperboloid in $\mathbb{R}^3$ with the metric induced by the Minkowski metric on $R^3$. Define $O^+(2,1)$ to be the subgroup of $GL(3, \mathbb{R})$ that preserves the Minkowski metric and takes the upper sheet of the hyperboloid to itself. Then $O^+(2, 1)$ acts by isometries on this model. $SO^+(2,1)$ is then the intersection of $O^+(2, 1)$ with $SL(3, \mathbb{R})$.
A second model is the upper half plane. Then $SL(2,\mathbb{R})$ acts by isometries on this model when each element of $SL(2,\mathbb{R})$ is viewed as a Möbius transformation. $PSL(2,\mathbb{R})$ is $SL(2,\mathbb{R})/\langle \pm I \rangle$ (i.e., $SL(2,\mathbb{R})$ mod the two element subgroup consisting of the identity and minus the identity).
How do I prove that these two groups are isomorphic as groups? Lie groups?