PSL(2,C) action on hyperbolic three-space

Statement

On the positive-definite Hermitian-matrix model of $\mathbb H^3$ , the formula

H\longmapsto AHA^\dagger,\qquad A\in SL(2,\mathbb C),

is isometric and has kernel $\{\pm I\}$ . It induces an isomorphism of real Lie groups

PSL(2,\mathbb C)_{\mathbb R} \xrightarrow{\;\sim\;} \operatorname{Isom}^+(\mathbb H^3).

Relation to the Lorentz model

Under the determinant-one hyperboloid model, this is the restriction of the spin-cover action to the sheet $q(v)=-1,\ t>0$ . Consequently,

\operatorname{Isom}^+(\mathbb H^3) \cong SO^+(1,3) \cong PSL(2,\mathbb C)_{\mathbb R}.

These isomorphisms are in the category of real Lie groups.

Boundary action and quotients

The action extends continuously to $\partial_\infty\mathbb H^3\cong\mathbb{CP}^1$ , where it is the Möbius action. Thus a discrete subgroup $\Gamma<PSL(2,\mathbb C)$ acts simultaneously on $\mathbb H^3$ and its sphere at infinity. The quotient $\Gamma\backslash\mathbb H^3$ is a hyperbolic orbifold, and is a manifold when $\Gamma$ acts freely. Arithmetic choices of $\Gamma$ connect this geometry with automorphic forms and representation theory.

References

John G. Ratcliffe, Foundations of Hyperbolic Manifolds, 3rd ed., Springer, 2019, §§3.2–3.3 and 4.5. Publisher record.
Alan F. Beardon, The Geometry of Discrete Groups, Springer, 1983, Chapters 4 and 7. Publisher record.