1

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!

Kyor
  • 61

2 Answers2

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.

Dietrich Burde
  • 130,978
  • I saw that $PSL(2,\mathbb{C})$ acts on the boundary of the upper half space by $$\left(\begin{array}{cc}a&b\c&d\end{array}\right).z=\frac{az+b}{cz+d}$$ and it can be extended to interior of $\mathbb{H}^3$. How can it be extended? – Kyor May 28 '14 at 02:16
  • I Read about the Poincare extension, for $Isom^+(\mathbb{H}^3)$ But I really don´t know what you mean when you wrote $Isom(\mathbb{H}^3)\cong PSL(2,\mathbb{C})\rtimes \mathbb{Z}/2\mathbb{Z}$. What is $\rtimes$? – Kyor May 28 '14 at 23:59
  • This denotes the semi-direct product. – Dietrich Burde May 29 '14 at 15:59
1

enter image description here

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.

Alan
  • 2,239
  • 13
  • 15