On quasiconformal groups

Recognizing constant curvature discrete groups in dimension 3

“…Conclusion (1) is one of the main conclusions of both [26] and [30]. Therefore we may assume that the action of G on S 2 is conformal.…”

“…(See Section 8.) Furthermore, among all possible topological conjugations of the action yielding a round 2-sphere as underlying space, we could use Mostow's rigidity theorem [22] and the Sullivan-Tukia theorem [26] [30] to show that there is at most one possible conjugation, up to quasiconformal homeomorphism, which makes the action uniformly quasiconformal in the classical sense. Among the uncountably many possible quasiconformal structures on S 2 , therefore, we are required to find exactly the right one.…”

“…Tukia [30] and Cannon and Cooper [8] have given an earlier 3-dimensional characterization of the groups of Theorem 2.3.1.…”

“…To that end, we assume that G is a group with a locally finite Cayley graph Γ quasi-isometric with hyperbolic 3-space H 3 . (It follows from Tukia [30], or as a special case of Cannon-Cooper [8], that G acts isometrically, properly discontinuously, and cocompactly on H 3 , but we shall not make use of that fact. This paper will, in the end, supply another (difficult) proof of that fact.)…”

“…Theorem 2.1.2 ([26] and [30]). In order that a group G act geometrically on H 3 , it is necessary and sufficient that G act discretely and uniformly quasiconformally on the 2-sphere S 2 .…”