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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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