Action of Möbius transformations on homeomorphisms: Stability and rigidity
The author studies the action of the group of Möbius transformations on the universal Teichmüller space (defined as the set of all normalized homeomorphisms of the circle at infinity of the hyperbolic plane with the appropriate topology) and its analogues in higher-dimensional hyperbolic spaces (here e.g. instead of circle, he considers spheres at infinity). He extends the characterization of quasisymmetric homeomorphisms to a characterization of quasiconformal homeomorphisms of the sphere at infinity. A homeomorphism \(f:S^1\to S^1\) is called normalized if it fixes three points \(a,b,c\in S^1\), where \(S^1\) is the unit circle of the complex plane \({\mathbf C}\). \(\text{id}:S^1\to S^1\) is considered the distinguished point of the set \(N\) of all normalized homeomorphisms and will denoted by \({\mathbf O}\). Let \(H^n_F\) be one of the hyperbolic spaces over the fields \({\mathbf F}={\mathbf R}\), \({\mathbf C}\), \({\mathbf H}\) of real numbers, complex numbers, quaternions or over the algebra \({\mathbf F}={\mathbf O}\) of Cayley numbers (in this case \(n=2\)). In particular, \(H^n_R=H^n\). Let \(S\) be the sphere at infinity of \(H^n_F\) and let \(H\) be the group of all homeomorphisms \(S\to S\). The group of isometries of \(H^n_f\) acts on \(S\) by homeomorphisms usually called Möbius transformations. This group is denoted by Möb. The first three sections of this paper are restricted to the case \(H^2\). Let \(H\) be the group of orientation-preserving homeomorphisms \(f:S^1\to S^1\) and Möb its subgroup of Möbius transformations. In section 1, he considers the points \(f\in H\) which are stable (i.e. \(f\neq 0\) and its Möb-orbit is closed) or semi-stable (i.e. the closure of its Möb-orbit does not contain \({\mathbf O}\)), while in Section 2, the points of \(H\) fixed by some Fuchsian group, which is equivalent with considering homeomorphisms conjugating one Fuchsian group to another one. In section 3, he establishes the following pointwise non-rigidity theorem: ``For any torsion-free Fuchsian group \(\Gamma\) there exists a homeomorphism \(f:S^1\to S^1\) with the following properties: (i) \(f\) conjugates \(\Gamma\) to some other Fuchsian group \(\Gamma'\); (ii) \(f\) is not a Möbius transformation; (iii) \(f\) is differentiable with a non-zero derivative at any point which is not a conical limit point of \(\Gamma\); (iv) \(f\) is Lipschitz differentiable with a non-zero derivative at any point which is not a horospherical limit point of \(\Gamma\). We recall that a point \(z\in S^1\) is said to be a conical limit point of \(\Gamma\) if for some hyperbolic line \(L\) ending at \(z\) and some point \(x\in D\) (where \(D\) is the open unit disc), there exists a sequence of elements \(g_1,g_2,\dots\in\Gamma\) such that \(\lim_{i\to\infty} g_i(x)=z\) and \(d[g_i(x),L]\leq C\) for some constant \(C\), where \(d(.,.)\) is the hyperbolic distance on \(D\). We precise that a function \(f:\mathbb{R}\to\mathbb{R}\) is Lipschitz differentiable of order \(k\leq 1\) at a point \(x\in \mathbb{R}\) if there is a polynomial \(p(y)\) of degree \(\leq k\) such that \({|f(x+y)-p(y)|\over|y|^{k+1}}\) is bounded when \(y\to0\). We precise that a point \(z\in S^1\) is called a horoscopical limit point of \(\Gamma\) if for some point \(x\in D\) there exists a sequence of elements \(g_1,g_2,\dots\in\Gamma\) such that for any horodisk \(B\) based at \(z\) the points \(g_i(x)\) belong to \(B\) for all sufficiently large \(i\). In section 4, he explains how to extend this theory to \(H^n\). Finally, in section 5, he deals with the other hyperbolic spaces \(H^n_{\mathbf C}\), \(H^n_{\mathbf H}\), \(H^n_{\mathbf O}\).
scientific article; zbMATH DE number 881246
scientific article