Differentiability Of Integrable Measurable Cocycles Between Nilpotent Groups

2015/09/29 by Michael A. Cantrell, Michael Cantrell, Cantrell, Michael · 1 citation
Mathematics · #20E99 #20F65 #28D15 #37A20 #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Group Theory (math.GR) #math.DS #math.GR #msc:20E99 #msc:20F65 #msc:28D15 #msc:37A20

paper · pdf · doi:10.48550/arxiv.1509.08966

New version corrects a minor mathematical error in the statement and use of the Guivarc'h lemma. The corrections significantly simplify the proofs, particularly section 4

openalex publication_date 2015/09/29 · arxiv created 2016/07/29 · arxiv updated 2016/08/02 · openalex created_date 2022/08/14 · openalex updated_date 2026/07/28

Abstract

We prove an analog for integrable measurable cocycles of Pansu's differentiation theorem for Lipschitz maps between Carnot-Carathéodory spaces. This yields an alternative, ergodic theoretic proof of Pansu's quasi-isometric rigidity theorem for nilpotent groups, answers a question of Tim Austin regarding integrable measure equivalence between nilpotent groups, and gives an independent proof and strengthening of Austin's result that integrable measure equivalent nilpotent groups have bi-Lipschitz asymptotic cones. Our main tools are a nilpotent-valued cocycle ergodic theorem and a Poincaré recurrence lemma for nilpotent groups.

Cited by

• L^p measure equivalence of nilpotent groups

Related

• Integrable measure equivalence for groups of polynomial growth
• On the converse of Pansu's Theorem
• Carnot rectifiability and Alberti representations
• L^p measure equivalence of nilpotent groups
• Porosity, Differentiability and Pansu's Theorem