Quasisymmetric groups

Abstract

One of the first problems in the theory of quasisymmetric and convergence groups was to investigate whether every quasisymmetric group that acts on the sphere S n \textbf {S}^{n} , n > 0 n>0 , is a quasisymmetric conjugate of a Möbius group that acts on S n \textbf {S}^{n} . This was shown to be true for n = 2 n=2 by Sullivan and Tukia, and it was shown to be wrong for n > 2 n>2 by Tukia. It also follows from the work of Martin and of Freedman and Skora. In this paper we settle the case of n = 1 n=1 by showing that any K K -quasisymmetric group is K 1 K_1 -quasisymmetrically conjugated to a Möbius group. The constant K 1 K_1 is a function K K .

Big mapping class groups: an overview

“…A hyperbolic Riemann surface is a complex 1-manifold whose universal cover is isomorphic to the unit disk. In [86], Markovic proved a generalization of Nielsen realization for analytically-infinite Riemann surfaces, which we state as follows:…”

Cross-ratio distortion and Douady–Earle extension: III. How to control the dilatation near the origin

“…It is proved in [5] that the maximal dilatation K(Φ(f )) on the unit disk D depends on ||f || cr in a linear fashion. Using the expressions ( 9)-( 16), Markovic developed a criterion (Lemma 3.6, [9]) for a family of circle homeomorphisms f to have a uniform upper bound for the maximal dilatations of their Douady-Earle extensions on a uniform neighborhood of the origin. There are two goals for this paper: one is to introduce some explicit conditions on f such that f satisfies Markovic's criterion; the other goal is to show some explicit conditions on f that do not lead to any upper bound for the maximal dilatation of Φ(f ) on any neighborhood of the origin.…”

Grupos de difeomorfismos del círculo

“…Observación. De acuerdo a un resultado de Markovic [131], bajo las hipótesis precedentes el grupo Γ es quasi-simétricamente conjugado a un subgrupo de PSL(2, R). c]) = ∞ para todo a < b < c < a (observe que la corriente de Liouville satisface estas propiedades).…”