Isometries and inner product in Hyperbolic SPACE (H3)

Question

Well, the title says what I need to know/understand, when I studied upper half plane I remember that isometries are Mobius transformations (If I am not wrong), now I have no clue about it. Thanks for your help!

Answer 1

With the upper-half space model of hyperbolic $3$-space, it is easy to see that the orientation-preserving isometry group of $\mathbb{H}^3$ is $PSL(2,\mathbb{C})$, as was first observed by Poincare. The Moebius transformations $$ z\mapsto \frac{az+b}{cz+d} $$ from $\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}$ extend to an isometry of $\mathbb{H}^3$. Furthermore $PSL(2,\mathbb{\mathbb{C}})$ is of index $2$ in the full isometry group $Isom(\mathbb{H}^3)\cong PSL(2,\mathbb{C})\rtimes \mathbb{Z}/2\mathbb{Z}$, where the nontrivial element of $\mathbb{Z}/2\mathbb{Z}$ acts by complex conjugation on $PSL(2,\mathbb{C})$. A very good reference is the book Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory, by Juergen Elstrodt, Fritz Grunewald and Jens Mennicke.

Answer 2

The action of an element of SL(2,C) on the upper half space model of hyperbolic 3- space. The Moebius transformation A , gives rise to an axis(A) (shown in red) a geodesic in the space. There are two fixed points (shown in black) -- a hyperbolic or loxodromic element. hyperbolic planes are translated along the axis into hyperbolic planes , (either hemispheres or planes), in the upper half space model these are equivalent. The model is due to Poincare , from his first papers on Kleinian groups.