Lipschitz and biLipschitz Maps on Carnot Groups

2010/03/19 by William Meyerson, Meyerson, William
Mathematics · #43A80 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #math.MG #msc:43A80

paper · pdf · doi:10.48550/arxiv.1003.3723

openalex publication_date 2010/03/19 · arxiv created 2012/09/07 · arxiv updated 2012/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose A is an open subset of a Carnot group G, where G has a discrete analogue, and H is another Carnot group. We show that a Lipschitz function from A to H whose image has positive Hausdorff measure in the appropriate dimension is biLipschitz on a subset of A of positive Hausdorff measure. We then construct Lipschitz maps from open sets in Carnot groups to Euclidean space that do not decrease dimension. Finally, we discuss two counterexamples to explain why Carnot group structure is necessary for these results.

Citations

• Metriques de Carnot-Caratheodory et Quasiisometries des Espaces Symetriques de rang un

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