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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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References:
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