Sobolev mappings between nonrigid Carnot groups

TL;DR

This paper proves a rigidity conjecture for nonrigid Carnot groups by showing quasisymmetric homeomorphisms preserve specific foliation structures, extending understanding of geometric mappings in these complex groups.

Contribution

It demonstrates that quasisymmetric homeomorphisms in nonrigid Carnot groups are reducible, preserving coset foliations, and confirms the quasisymmetric rigidity conjecture for these groups.

Findings

Quasisymmetric homeomorphisms preserve coset foliations in nonrigid Carnot groups.

The quasisymmetric rigidity conjecture is proven for nonrigid Carnot groups.

The result excludes the case of R^n and Heisenberg groups where the assertion fails.

Abstract

We consider mappings between Carnot groups. In this paper, which is a continuation of "Pansu pullback and rigidity of mappings between Carnot groups" (arXiv:2004.09271), we focus on Carnot groups which are nonrigid in the sense of Ottazzi-Warhurst. We show that quasisymmetric homeomorphisms are reducible in the sense that they preserve a special type of coset foliation, unless the group is isomorphic to R^n or a real or complex Heisenberg group (where the assertion fails). We use this to prove the quasisymmetric rigidity conjecture for such groups. The starting point of the proof is the pullback theorem established our previous paper.