×

Strange actions of groups on spheres. II. (English) Zbl 0666.57028

Holomorphic functions and moduli II, Proc. Workshop, Berkeley/Calif. 1986, Publ., Math. Sci. Res. Inst. 11, 41-57 (1988).
[For the entire collection see Zbl 0646.00005.]
This paper continues the discussion of Part I [J. Differ. Geom. 25, 75–98 (1987; Zbl 0588.57024); see also F. W. Gehring and G. Martin, Proc. Lond. Math. Soc. 55, 331–358 (1987; Zbl 0628.30027)] about admissible group actions on higher dimensional spheres (topological analogues of Schottky groups). As in the previous paper for dimension three, the authors not only construct such actions but analyze their compatibility with various structures on spheres. Each action described here has the property that each homeomorphism (group element) is individually topologically conjugate to a Möbius transformation, but the group action (except possibly the uniformly quasiconformal action \(F_ r\times S^ 3\to S^ 3\) of § 4) is not topologically conjugate to a conformal action. Particularly, for \(n\geq 4\) and sufficiently large \(r=r(n)\), the authors construct an admissible action \(\psi_ n: (F_ r\rtimes\mathbb Z_{2r})\times S^ n\to S^ n\) which is smooth and uniformly quasiconformal but not conjugate to a conformal action.

MSC:

57S30 Discontinuous groups of transformations
57N45 Flatness and tameness of topological manifolds
57S25 Groups acting on specific manifolds
30C62 Quasiconformal mappings in the complex plane
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)