Reference for the proof that Möbius transformations extend to isometries of hyperbolic 3-space

Question

Consider the group $\operatorname{PSL}(2,\mathbb C)$ acting by Möbius transformations of the Riemann sphere. It is known that this action can be extended to an action on the unit ball which is isometric with respect to the hyperbolic metric. To prove this you write each Möbius transformation as a product of inversions in spheres and show that each of these act by isometries.

My question: can anyone point out a reference where this is done rigorously and explicitly (or explain the computation)? Everywhere I read the proof that inversions in spheres act by isometries on the unit ball with the hyperbolic metric is left as an "exercise" which I cannot do. I am teaching a class on this but need to understand myself...

Answer 1

Probably, you can find a discussion of this in Thurston's notes on hyperbolic 3-manifolds, or maybe some of the expositions by his students.

However, what you are asking for is actually pretty simple: Consider the vector space $V$ of $2$-by-$2$ Hermitian symmetric matrices, thus, $A\in M_2(\mathbb{C})$ lies in $V$ if and only if $A = A^*$, where $A^*$ denotes the conjugate transpose of $A$. The group $\mathrm{SL}(2,\mathbb{C})$ acts on $V$ by $g\cdot A = g A g^*$, and it preserves the (real-valued) quadratic form $Q$ defined by $Q(A) = \det(A)$. Since $$ Q\left(\begin{pmatrix}a+b & z\\ \bar z & a-b\end{pmatrix}\right) = a^2 - b^2 - z\bar z, $$ the type of $Q$ is $(1,3)$.

Note that this action of $\mathrm{SL}(2,\mathbb{C})$ on $V\simeq\mathbb{R}^{1,3}$ is not effective, since $\pm I_2\in\mathrm{SL}(2,\mathbb{C})$ acts trivially. However, these are the only two elements that act trivially, so this is a faithful representation of $\mathrm{PSL}(2,\mathbb{C}) = \mathrm{SL}(2,\mathbb{C})/\{\pm I_2\}$ on $V$. By dimension count, this representation of $\mathrm{PSL}(2,\mathbb{C})$ is the identity component of $\mathrm{O}(Q) \simeq \mathrm{O}(1,3)$.

Now, one can identify hyperbolic $3$-space $H^3$ with the set of matrices in $V$ of determinant $1$ and positive trace, which is the $\mathrm{PSL}(2,\mathbb{C})$-orbit of $I_2\in V$. (Note that the $\mathrm{PSL}(2,\mathbb{C})$-stabilizer of $I_2\in V$ is $\mathrm{SU}(2)/\{\pm I_2\}\simeq \mathrm{SO}(3)$.) Alternatively, one can identify hyperbolic $3$-space with the space of (real) $1$-dimensional subspaces of $V$ on which $Q$ is positive definite. In this way, $H^3$ is an open subset of $\mathbb{RP}^3 = \mathbb{P}(V)$, which is an orbit of $\mathrm{PSL}(2,\mathbb{C})$ in its natural action on $\mathbb{P}(V)$.

Now consider the set $N$ of $1$-dimensional subspaces of $V$ on which $Q$ vanishes. This is the boundary of hyperbolic space in $\mathbb{RP}^3$. It is not difficult to show that $\ell\in N$ is spanned by a matrix of the form $$ \begin{pmatrix}p\bar p & p\bar q\\ q\bar p & q\bar q\end{pmatrix}. $$ While this does not determine $(p,q)\in\mathbb{C}$ uniquely, it does determine $[p,q]\in\mathbb{CP}^1$, and, vice versa, $[p,q]\in\mathbb{CP}^1$ determines $\ell\in N$.

Thus, $\mathrm{PSL}(2,\mathbb{C})$ acts smoothly and effectively on the space of lines in $\mathbb{RP}^3$ on which $Q$ is non-negative, which is a closed $3$-ball in $\mathbb{RP}^3$ so that the action in the interior is the (identity component of the) isometry group of $H^3$ and the action on the boundary is the standard action of $\mathrm{PSL}(2,\mathbb{C})$ on $\mathbb{CP}^1$, i.e., the Riemann sphere.

Answer 2

This holds more generally in higher dimension too, and I needed similar references about this basic fact of the so-called AdS/CFT correspondence: global conformal maps of the sphere $S^n$ are in one-to-one correspondence with isometries of the ball $B^{n+1}$ with the hyperbolic metric.

The most useful references I found were:

J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149 (Springer, New York, 2006). See in particular Theorem 4.5.2.

and

J. B.Wilker, "Inversive geometry". In: The Geometric Vein, Eds. C. Davis, B. Grünbaum and F. A. Sherk, pp. 379–442 (Springer, New York-Berlin, 1981).

The nice thing also, is similar statements hold over the $p$-adics. See my review article

"Towards three-dimensional conformal probability", p-Adic Numbers Ultrametric Anal. Appl. 10 (2018), no. 4, 233-252

or a talk I gave which is on youtube.

BTW, I learned of Ratcliffe's book from Ryan Budney's answer to the related question

conformal diffeomorphism of sphere

Answer 3

MR0725161 Ahlfors, Lars V. Möbius transformations in several dimensions, Minneapolis, MN, 1981.

Answer 4

Another reference which takes a very explicit point of view (i.e. entirely matrix calculations and explicit circle inversions) is Beardon's The Geometry of Discrete Groups, which deduces the invariance in $n$ dimensions from the following useful formula: if $\sigma$ is the reflection in the Euclidean sphere of radius $r$ and centre $a$, then we can estimate the derivative of $ \sigma $ by

$$ \frac{|\sigma(y) - \sigma(x)|}{|y-x|} = \frac{r^2}{|x-a||y-a|} $$

(this is formula 3.1.5 on p.26). Observe now that the extension $\tilde{\phi}$ to $\mathbb{R}^{n+1} $ of any inversion $\phi$ in $ \mathbb{R}^n$ is a reflection in some circle of radius $r$ centred on $ a $ with $a_{n+1} = 0 $, and explicitly writing down the formula for circle inversion gives that the $(n+1)$th coordinate of $\tilde{\phi}(x) $ is

$$ \frac{r^2 x_{n+1}}{|x-a|^2}. $$

But this formula is almost $ x_{n+1} $ times the first displayed formula above, and this similarity suggests

\begin{align*} \frac{|\tilde{\phi}(y)-\tilde{\phi}(x)|^2}{\tilde{\phi}(x)_{n+1} \tilde{\phi}(y)_{n+1}} &= \frac{r^4 |y-x|^2}{|x-a|^2|y-a|^2 \frac{r^2 x_{n+1}}{|x-a|^2} \frac{r^2 y_{n+1}}{|y-a|^2}}\\ &= \frac{|y-x|^2}{x_{n+1} y_{n+1}} \end{align*}

i.e. the form $ |y-x|^2/x_{n+1} y_{n+1} $ is invariant under Poincare extensions of sphere inversions. The same is true for extensions of plane reflections, and since Poincare extension is a homomorphism we get that the extension of every Moebius transformation preserves that form; but the form is just the hyperbolic metric:

$$ \sum_{k=0}^{N} \sqrt{\frac{|\gamma(t^k) - \gamma(t^{k-1})|^2}{\gamma(t^k)_{n+1} \gamma(t^{k-1})_{n+1}}} \to \int \frac{|d\gamma(t)|}{\gamma(t)_{n+1}} $$

(where I have written an approximation to the length of the curve $ \gamma $ by taking a partition $ t^0, \dots, t^N $: the function under the sum is clearly the square root of the invariant form above, and it approaches the integral at the right where the integrand is the usual expression for the hyperbolic metric $|dx|/x_{n+1} $).

This calculation is found on pp.34--35 of Beardon (I have added a very small amount of detail). At this point of the book he has only done the structure of Moebius maps as products of reflections, i.e. it is very hands-on and requires essentially no "heavy" machinery. The downside of this approach is that the result appears magically out of thin air as opposed to other appproaches which come from symmetric spaces and Lie groups, e.g. showing that $ \mathrm{PSL}(2,\mathbb{C}) $ is naturally isomorphic to $ O(2,1) $ as a Lie group and hence admits an action by isometries on $ \mathbb{H}^3 $ which you then show is the usual action by fractional linear transformations.