Global contact and quasiconformal mappings of Carnot groups

Abstract

We show that globally defined quasiconformal mappings of rigid Carnot groups are affine, but that globally defined contact mappings of rigid Carnot groups need not be quasiconformal, and a fortiori not affine.

The contact mappings of a flat $(2,3,5)$-distribution

“…The recent paper [13] proves this in the case of nonrigid Carnot groups. In the case of rigid groups, the conjecture would be verified if it were true that all quasiconformal mappings were C ∞ -smooth (once coupled with the results of [9]). All geometric mappings such as bi-Lipschitz and quasiconformal mappings must be weakly contact.…”

Sobolev mappings between nonrigid Carnot groups

“…the differential Df preserves the horizontal subbundle V 1 ⊂ T G. The study of contact diffeomorphisms has a long and fascinating history intertwined with the theory of Lie pseudogroups, overdetermined systems, and G-structures (in the sense of E. Cartan); the literature extends back to the 19th century, with major contributions in 1900-10 by Cartan, in 1955-70 by Kuranishi, Singer, Sternberg, Guillemin, Quillen, and Tanaka (among many others); there has been a resurgence of interest in recent decades, coming from new connections with geometric group theory and quasiconformal mappings. The literature on this topic is substantial, so we will mention just a few points which are directly relevant to our setting, and refer the interested reader to [SS65,Tan70,CO15] for references and more discussion. The contact condition is a nonlinear system of PDEs which is formally overdetermined except when G is the Engel group or a product H k × R ℓ for some k, ℓ ≥ 0, and hence one expects some form of rigidity in the generic case; here H k denotes the k th Heisenberg group.…”

“…See Theorem 3.1 for a more precise statement. Note that for rigid groups Conjecture 1.4 would follow from the Regularity Conjecture 1.1 and [CO15].…”