zbMATH — the first resource for mathematics

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.

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
Full Text: DOI EuDML
[1] Agard, S.: A geometric proof of Mostow’s rigidity theorem for groups of divergence type. Acta Math.151, 231-252 (1983) · Zbl 0532.30038 · doi:10.1007/BF02393208
[2] Agard, S.: Elementary properties of Möbius transformations inR n with applications to rigidity theory. University of Minnesota, Math. Rep. (mimeographed notes) No. 82-100 (1982)
[3] Agard, S.: Remarks on the boundary mapping for a Fuchsian group. (To appear) · Zbl 0588.30049
[4] Ahlfors, L.V.: Möbius transformations in several dimensions. Mimeographed lecture notes at the University of Minnesota 1981 · Zbl 0517.30001
[5] Apanasov, B.N.: Geometrically finite groups of spatial transformations. Sib. Math. J. (Russian)23, (6) 16-27 (1982) · Zbl 0519.30038
[6] Beardon, A.F., Maskit, B.: Limit points of Kleinian groups and finite sided fundamental polyhedra. Acta Math.132, 1-12 (1974) · Zbl 0277.30017 · doi:10.1007/BF02392106
[7] Federer, H.: Geometric measure theory. New York: Springer 1969 · Zbl 0176.00801
[8] Garland, H., Raghunathan, M.S.: Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups. Ann. Math.92, 279-326 (1970) · Zbl 0206.03603 · doi:10.2307/1970838
[9] Gottschalk, W.H., Hedlund, G.A.: Topological dynamics. AMS Colloquium Publications XXXVI, Am. Math. Soc. 1955 · Zbl 0067.15204
[10] Lehto, O., Virtanen, K.I.: Quasiconformal mappings in the plane. Berlin-Heidelberg-New York: Springer 1973 · Zbl 0267.30016
[11] Mostow, G.D.: Strong rigidity of locally symmetric spaces. Ann. Math. Stud., vol. 78. Princeton: Princeton University Press 1973 · Zbl 0265.53039
[12] Näätänen, M.: Maps with continuous characteristics as a subclass of quasiconformal maps. Ann. Acad. Sci. Fenn., Ser. AI410, 1-28 (1967) · Zbl 0174.12701
[13] Selberg, A.: Recent developments in the theory of discontinuous groups of motions of symmetric spaces. In: Aubert, K.E., Ljunggren, W.L. (eds.), Proceedings of the 15th Scandinavian Congress, Oslo 1968. Lect. Notes Math., vol. 118, pp. 9-120. Berlin-Heidelberg-New York: Springer 1970
[14] Sullivan, D.: On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. In: Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook conference (I. Kra, B. Maskit, eds.), Ann. Math. Stud., vol. 97, pp. 465-496. Princeton: Princeton University Press 1981
[15] Tukia, P.: On two-dimensional quasiconformal groups. Ann. Acad. Sci. Fenn., Ser. AI5, 73-78 (1980) · Zbl 0411.30038
[16] Tukia, P.: On isomorphisms of geometrically finite Möbius groups. Inst. Hautes Études Sci. Publ. Math.61, 171-214 (1985) · Zbl 0572.30036 · doi:10.1007/BF02698805
[17] Tukia, P.: Rigidity theorems for Möbius groups. (To appear) · Zbl 0674.30038
[18] Tukia, P.: On quasiconformal groups. (To appear) · Zbl 0603.30026
[19] Wielenberg, N.J.: Discrete Möbius groups: Fundamental polyhedra and convergence. Am. J. Math.99, 861-877 (1977) · Zbl 0373.57024 · doi:10.2307/2373869
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.