Quasiregular mappings between equiregular sub-Riemannian manifolds

Abstract

In this paper, we provide an alternative approach to an expectation of Fässler et al [J. Geom. Anal. 2016] by showing that a metrically quasiregular mapping between two equiregular sub-Riemannian manifolds of homogeneous dimension $Q⩾2$ has a negligible branch set. One main new ingredient is to develop a suitable extension of the generalized Pansu differentiability theory, in spirit of earlier works by Margulis–Mostow, Karmanova, and Vodopyanov. Another new ingredient is to apply the theory of Sobolev spaces based on upper gradients developed by Heinonen, Koskela, Shanmugalingam, and Tyson to establish the necessary analytic foundations.