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-Caratheodory 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 Poincare recurrence lemma for nilpotent groups.