Equivalence of quasiregular mappings on subRiemannian manifolds via the Popp extension

TL;DR

This paper proves the equivalence of various definitions of quasiregular mappings on equiregular subRiemannian manifolds, introducing Popp extensions to unify the theory and answer open questions.

Contribution

It introduces Popp extensions to define and analyze quasiregularity, establishing their equivalence across different definitions in subRiemannian geometry.

Findings

All common definitions of quasiregular mappings are quantitatively equivalent.

1-quasiregular mappings are equivalent under all definitions.

Distortion estimates for Popp extensions are developed and analyzed.

Abstract

We show that all the common definitions of quasiregular mappings $f\colon M\to N$ between two equiregular subRiemannian manifolds of homogeneous dimension $Q\geq 2$ are quantitatively equivalent with precise dependences of the quasiregularity constants. As an immediate consequence, we obtain that if $f$ is $1$-quasiregular according to one of the definitions, then it is also $1$-quasiregular according to any other definition. In particular, this recovers a recent theorem of Capogna et al. on the equivalence of $1$-quasiconformal mappings. Our main results answer affirmatively a few open questions from the recent research. The main new ingredient in our proofs is the distortion estimates for particular local extensions of the horizontal metrics. These extensions are named "Popp extensions", and based on these extensions, we introduce a new natural and invariant definition of quasiregularity in the equiregular subRiemannian setting. The analysis on Popp extensions and on the implied distortion is also of independent interest.