Gromov hyperbolicity in the free quasiworld. I

TL;DR

This paper explores Gromov hyperbolicity in uniform domains, providing new characterizations and conditions for quasisymmetry and boundary behavior of quasiconformal mappings in Banach spaces.

Contribution

It offers a Gromov hyperbolic characterization of uniform domains and clarifies the necessity of the three-point condition for quasisymmetry in quasim"obius maps.

Findings

Affirmative answer to V"ais"al"a's open question under weaker assumptions

Three-point condition is necessary for quasisymmetry in bounded spaces

Partial answers to boundary behavior of quasiconformal mappings in Banach spaces

Abstract

With the aid of a Gromov hyperbolic characterization of uniform domains, we first give an affirmative answer to an open question arisen by V\"ais\"al\"a under weaker assumption. Next, we show that the three-point condition introduced by V\"ais\"al\"a is necessary to obtain quasisymmetry for quasim\"obius maps between bounded connected spaces in a quantitative way. Based on these two results, we investigate the boundary behavior of freely quasiconformal and quasihyperbolic mappings on uniform domains of Banach spaces and partially answer another question raised by V\"ais\"al\"a in different ways.