Contact lifts and Holder lifts to central extension of Carnot groups

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 γ-Holder lift when γ > n+1/n for step-n Carnot group.