Differentiability of integrable measurable cocycles between nilpotent groups (Q520761)

Summary: We prove an analog for integrable measurable cocycles of \textit{P. Pansu}'s differentiation theorem for Lipschitz maps between Carnot-Carathéodory spaces [Ann. Math. (2) 129, No. 1, 1-60 (1989; Zbl 0678.53042)]. 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 \textit{T. Austin}'s result that integrable measure equivalent nilpotent groups have bi-Lipschitz asymptotic cones [Groups Geom. Dyn. 10, No. 1, 117--154 (2016; Zbl 1376.20042)]. Our main tools are a nilpotent-valued cocycle ergodic theorem and a Poincaré recurrence lemma for nilpotent groups.