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A rigidity theorem for Möbius groups. (English) Zbl 0674.30038
Let G be a group of Möbius transformations of $$S^ n$$ and $$A\subset S^ n$$ a G-invariant set. A finite measure $$\mu$$ on A is a G-measure of dimension d if $$\mu (gE)=\int_{E}| g'(x)|^ d d\mu$$ for all $$g\in G$$ and any measurable $$E\subset A$$ (here $$| g'(x)|$$ is the norm of the derivative). D. Sullivan [Publ. Math. Inst. Hautes Étud. Sci. 50, 171-202 (1979; Zbl 0439.30034)] has shown that there is always a non-trivial G-measure supported by the limit set L(G) of G and that the action of G on L(G)$$\times L(G)$$ which sends (x,y) to (g(x),g(y)) is ergodic if G is geometrically finite.
Suppose that H is another Möbius group of $$S^ n$$, $$\nu$$ a H-measure on a H-invariant set B. Suppose that there is a homomorphism $$\phi$$ : $$G\to H$$ and a measurable map f: $$A\to B$$ which maps measurable sets onto measurable sets and induces $$\phi$$ in the sense that $$fg=\phi (g)f$$ on A. By the main theorem of the paper there is the following dichotomy for the map f whenever the action of G is ergodic on $$A\times A$$ and the dimension $$d>0:$$ Either f is a.e. the restriction of a Möbius transformation or f is singular in the sense that it maps a set of full $$\mu$$-measure onto a $$\nu$$-nullset. For geometrically finite groups of the first kind this theorem gives an alternative proof of Mostow’s rigidity theorem. Another theorem of the paper says that if the action on $$A\times A$$ is ergodic, then any map f: $$A\to B$$ (measurable or not) which induces a homomorphism $$\phi$$ : $$G\to H$$ must be either injective outside a nullset or then f is very badly non-injective: no two open sets of positive $$\mu$$-measure can have disjoint images. In addition the paper contains several examples of situations to which these theorems apply.
Reviewer: P.Tukia

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 22E99 Lie groups
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