Differentiability of integrable measurable cocycles between nilpotent groups
Michael Cantrell
University of Illinois at Chicago, USA
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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.
Cite this article
Michael Cantrell, Differentiability of integrable measurable cocycles between nilpotent groups. Comment. Math. Helv. 92 (2017), no. 1, pp. 185–213
DOI 10.4171/CMH/410