Differentiability Of Integrable Measurable Cocycles Between Nilpotent Groups

TL;DR

This paper extends Pansu's differentiation theorem to integrable measurable cocycles between nilpotent groups, providing new proofs and insights into their geometric and ergodic properties, including measure equivalence and asymptotic cone structures.

Contribution

It introduces an ergodic theoretic approach to Pansu's theorem for integrable cocycles, offering new proofs and strengthening results on measure equivalence and asymptotic cones of nilpotent groups.

Findings

Proves an analog of Pansu's differentiation theorem for integrable cocycles.

Provides an ergodic theoretic proof of Pansu's quasi-isometric rigidity for nilpotent groups.

Shows integrable measure equivalent nilpotent groups have bi-Lipschitz asymptotic cones.

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.