Gromov hyperbolicity in the free quasiworld. I

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.