Topological multiple recurrence for polynomial configurations in nilpotent groups

We establish a general multiple recurrence theorem for an action of a nilpotent group by homeomorphisms of a compact space. This theorem can be viewed as a nilpotent version of our recent polynomial Hales-Jewett theorem ([BL2]) and contains nilpotent extensions of many known “abelian” results as special cases. 0. Introduction 0.1. The celebrated van der Waerden theorem on arithmetic progressions, published in 1927 ([vdW]) states that if the set of integers is partitioned into finitely many classes then at least one of the classes contains arbitrarily long arithmetic progressions. Few years later Grünwald (=Gallai) obtained the following multidimensional extension of van der Waerden’s theorem (see [R], p. 123). 0.2. Theorem. Let d ∈ N. For any finite coloring of Z and any finite set E ⊂ Z there exist v ∈ Z and n ∈ N such that the set v + nE = {