Quasiconformal and affine groups. (English) Zbl 0668.53004

Martin, Gaven J.

J. Differ. Geom. 29, No. 2, 427-448 (1989).

Suppose that G is a discrete group of diffeomorphisms acting on the unit sphere \(S^ n\) of \({\mathbb{R}}^{n+1}\). G is called admissible if it contains an infinite cyclic virtually central subgroup. (Recall that a subgroup H of G is called virtually central if the commutator [G,H] is finite.) The main result of the paper is that if G is admissible and has uniformly bounded distortion, then G is conjugate, by a self- homeomorphism of \(S^ n\) with bounded distortion, to a conformal group of \(S^ n\). It is shown that, under some restrictions, the parabolic and loxodromic elements of a uniformly quasiconformal group are quasiconformally conjugate to conformal transformations and good bounds are obtained on the dilatation of the conjugating mapping.

Reviewer: B.Csikós

MSC:

53A30	Conformal differential geometry (MSC2010)
53C20	Global Riemannian geometry, including pinching

Keywords:

discrete group of diffeomorphisms; infinite cyclic virtually central subgroup; uniformly bounded distortion; conformal group; parabolic and loxodromic elements; quasiconformal group; dilatation