Intrinsically Lipschitz graphs on semidirect products of groups

TL;DR

This paper explores the properties of intrinsically Lipschitz graphs within semidirect product groups, providing equivalent conditions and adapting concepts from Carnot groups to a broader context without intrinsic dilations.

Contribution

It extends the theory of intrinsically Lipschitz maps to semidirect product groups, replacing intrinsic dilation with projection map properties.

Findings

Established equivalent conditions for intrinsically Lipschitz maps in semidirect products.

Demonstrated that Lipschitz projections suffice in the absence of intrinsic dilations.

Extended the framework of intrinsically Lipschitz graphs beyond Carnot groups.

Abstract

In the metric spaces, we give some equivalent condition of intrinsically Lipschitz maps introduce by Franchi, Serapioni and Serra Cassano in subRiemannian Carnot groups. Unlike what happens in the Carnot groups, in our context intrinsic dilation do not exist but we can prove the same results using the Lipschitz property of the projection maps.