An Approach to Studying Quasiconformal Mappings on Generalized Grushin Planes

Abstract

We demonstrate that the complex plane and a class of generalized Grushin planes $G_r$, where $r$ is a function satisfying specific requirements, are quasisymmetrically equivalent. Then using conjugation we are able to develop an analytic definition of quasisymmetry for homeomorphisms on $G_r$ spaces. In the last section we show our analytic definition of quasisymmetry is consistent with earlier notions of conformal mappings on the Grushin plane. This leads to several characterizations of conformal mappings on the generalized Grushin planes.