Differentiability Of Integrable Measurable Cocycles Between Nilpotent Groups
Abstract
We prove an analog for integrable measurable cocycles of Pansu's differentiation theorem for Lipschitz maps between Carnot-Carath\'eodory 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 Poincar\'e recurrence lemma for nilpotent groups.
Keywords
Cite
@article{arxiv.1509.08966,
title = {Differentiability Of Integrable Measurable Cocycles Between Nilpotent Groups},
author = {Michael Cantrell},
journal= {arXiv preprint arXiv:1509.08966},
year = {2016}
}
Comments
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