Snippets

Isomorphic groups via hyperbolic geometry

$\mathrm{PSO}(1, 2)$ is the isometry group of hyperbolic 2-space under the hyperboloid model; $\mathrm{PSL}(2,\mathbb{R})$ is the isometry group of the upper half-plane; $\mathrm{PSU}(1,1)$ is the isometry group of the Poincaré unit disk. Since these are in fact models for the same space, all of these Lie groups are isomorphic.

Question: is there a proof of this isomorphism that doesn’t proceed through this identification as isometry groups? (Probably, via the Lie algebras for instance, but I’m a little too lazy to try and work it out at the moment.)