Conformal mappings on the Grushin plane

We study conformal mappings in the Grushin plane and provide a number of their characterizations in terms of the Sobolev mappings and their geometry. Furthermore, we connect conformality on the Grushin plane with conformality on the complex plane by using the Meyerson map. Among applications we discuss admissible curves and length-distortion estimates in the Grushin plane, as well as the Carathéodory extension theorem.