Contact lifts and Holder lifts to central extension of Carnot groups

TL;DR

This paper investigates the existence and properties of liftings of maps between Carnot groups using central extensions, establishing conditions for contact and Holder lifts, and revealing rigidity phenomena for quasiconformal maps.

Contribution

It introduces new existence results for contact lifts of Lipschitz and Sobolev maps, and provides a rigidity theorem for quasiconformal maps in Carnot groups.

Findings

Contact lifts exist for Lipschitz and Sobolev maps.

Quasiconformal maps with contact lifts are bi-Lipschitz.

Necessary criteria for Holder lifts when b3 > n+1/n.

Abstract

We consider the existence problem of lift F of a map f between Carnot group with different smoothness, where we use central extension to define lifting. Our main result is the existence of the contact lifts of Lipschitz and Sobolev maps and the rigidity result for the contact lift of quasiconformal maps: a quasiconformal map admits a contact lift then it is bi-Lipschitz. We also show a necessary criterion for the extension of {\gamma}-Holder lift when {\gamma} > n+1/n for step-n Carnot group.