Uniformly Quasisymmetric Groups

Abstract

An increasing homeomorphism / of the real line R onto itself is K-quasisymmetric if holds for all .v 6 R, t # 0. A K-quasisymmetric group is a group of K-quasisymmetric mappings under composition of functions.It is proved that if G is a K-quasisymmetric group, then there exists a quasisymmetric function / such that / " ' o G of is a group of linear functions.Pmc. London Math. Soc.(3). 51 (1985). 318 338. UNIFORMLY QUASISYMMETRIC GROUPS3 1 9 / / G is not cyclic and contains a function without fixed points, then f is uniquely determined up to a linear transformation and q(f) ^ K. REMARK 1. In general, there are a variety of choices for /. More details will be provided in the course of the proof. REMARK 2. If fg are qs, then q{f~l) and q{fog) are bounded by numbers depending on q(f) and q(g) only [3]; a linear function is 1-qs. Hence, if/ is qs and if Gj is a group of linear functions, then / o G t o/" 1 is a uniformly quasisymmetric group. REMARK 3. It will turn out that if G is not cyclic and if all functions of G have a fixed point, then all of them fix the same point (and no other point). REMARK 4. If G is not cyclic and contains a function without fixed points, then the equality q(f) = K is possible. This happens, for example, if G = / o Gj o / " 1 , where f(x) = x for x ^ 0, f(x) = Kx for x ^ 0, K ^ 1, and G { is the group of all translations.In §2 we prove preliminary results (Lemmas 1-5), and in §3 we prove Theorem 1 for cyclic groups. In §4 we outline the proof in the general case, and in § § 5 and 6 we prove the lemmas required to complete the proof of Theorem 1. I wish to thank Professor W. K. Hayman for several helpful suggestions. I am also grateful to the referee for his comments, and in particular for showing how the proof of Theorem 1 for non-cyclic groups can be considerably shortened by making use of the compactness properties of quasisymmetric functions. Preliminary resultsWe recall some distortion properties of qs functions. If L,-(x) = a,x + 6,, with a { > 0 and / = 1,2, if / is qs and g(x) = -/ ( -x), then [3]

Foliations, topology and geometry of 3-manifolds: R-covered foliations and transverse pseudo-Anosov flows

“…The 4 points in Z define an unique ideal quadrilateral in H 2 with ideal points in Z, which is regular if and only if C(Z) is equal to 1/2. For any homeomorphism f of Hi1,Hi2]. This means regular quadrilaterals do not get too distorted.…”

“…There is a rich theory of quasisymmetric maps [Hi1,Hi2,Le]. A group Γ acting on S 1 is uniformly quasisymmetric if there is K so that for any f ∈ Γ, then f is a K-quasisymmetric homeomorphism of S 1 .…”

Abelian and nondiscrete convergence groups on the circle

“…To prove Lemma 12, we may proceed as in [2, pp. 335-336] when proving the case G = G T of [2,Lemma 10]. The convergence properties required for Sublemma 1 in [2, p. 335] now follow from the fact that G is a convergence group.…”

Quasisymmetric groups

“…In particular, the following proposition is a subcollection of results proved in [14] and [16] that are going to be used in this paper.…”

“…It is a part of the definition of quasisymmetric (and quasiconformal maps in general) that they are sense-preserving. Originally, quasiconformal groups are defined to be sense-preserving (see [11], [14]). However, one can naturally extend this definition to sense-reversing maps.…”