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Measurable rigidity for Kleinian groups. (English) Zbl 1379.37090
Deformation and rigidity are very important topics in the theory of Kleinian groups. D. Sullivan [ Ann. Math. Stud. 97, 465–496 (1981; Zbl 0567.58015)] proved that there are no quasi-conformal deformations supported on limit sets, which is analogous to Mostow rigidity. Moreover, he proved a measurable rigidity theorem and P. Tukia [Invent. Math. 97, No. 2, 405–431 (1989; Zbl 0674.30038)] extended this rigidity in a more general setting.
Let \(G\) and \(H\) be Kleinian groups, and \(\mu_G\) and \(\mu_H\) conformal measures, where \(\mu_G\) has no atom. Suppose that at least one of the dimensions \(\mu_G\) and \(\mu_H\) is positive. Suppose that there is an essential injective, measurable and essentially directly measurable (i.e., the image of any measurable set outside some fixed \(\mu_G\)-null set is measurable) map \(f:\Lambda_G\to \Lambda_H\) which conjugates \(G\) to \(H\) almost everywhere. Then either there is a set \(A\) in \(\Lambda_G\) of full measure with \(\mu_H(f(A))=0\) or \(f\) coincides with the restriction of a conformal automorphism to \(\Lambda_G\) almost everywhere and the dimensions of \(\mu_G\) and \(\mu_H\) coincide.
However the existence of such map \(f\) is not evident. The authors prove that an equivariant map from the limit set of \(G\) to that of \(H\) is of divergence type.

MSC:
37F30 Quasiconformal methods and Teichm├╝ller theory, etc. (dynamical systems) (MSC2010)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
57M60 Group actions on manifolds and cell complexes in low dimensions
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