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Differentiability and rigidity of Möbius groups. (English) Zbl 0564.30033
Let G and H be groups of Möbius transformations of $$\bar R^ n$$, let $$j: G\to H$$ be a homomorphism and let $$A\subset \bar R^ n$$ be a G- invariant set. Suppose that $$f: A\to \bar R^ n$$ is a map such that $$fg(y)=j(g)f(y)$$ for $$y\in A$$ and $$g\in G$$ and that f is differentiable with a non-singular derivative at a radial point x of G (x is also called a conical limit point or a non-tangential limit point).
It is shown that f is, up to compositions with Möbius transformations, the restriction of an affine map, apart from a rather special set of circumstances and even then with at most one exceptional point. Furthermore, f is often a Möbius transformation, for instance if $$A=\bar R^ n$$ and no $$z\in \bar R^ n$$ is fixed by every $$g\in G.$$
The idea is the following. Suppose that $$g_ i(z)$$ radially approach x. Here $$g_ i\in G$$ and z is a point in the hyperbolic space whose boundary is $$\bar R^ n$$. Looking at x from the points $$g_ i(z)$$, f begins to look more and more like an affine map as $$i\to \infty$$. - Similar theorems consider differentiability at a limit point and the matrix dilatation at a radial or a limit point. The matrix dilatation is the generalization of the complex dilatation for $$n\geq 2$$. Indeed, if f is a quasiconformal homeomorphism of $$\bar R^ n$$ and if the matrix dilatation of f is approximately continuous at a radial point, or continuous at a limit point, then the above conclusions hold.
These theorems have relevance to Mostow’s rigidity theorem to which they allow to give new proofs.

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 58H15 Deformations of general structures on manifolds
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