By Ross Hewitt, Edwin Hewitt, Kenneth Ross

This booklet is a continuation of vol. I (Grundlehren vol. a hundred and fifteen, additionally to be had in softcover), and encompasses a specified remedy of a few vital components of harmonic research on compact and in the neighborhood compact abelian teams. From the studies: "This paintings goals at giving a monographic presentation of summary harmonic research, way more whole and accomplished than any booklet already present at the subject...in reference to each challenge taken care of the ebook bargains a many-sided outlook and leads as much as most up-to-date advancements. Carefull awareness is usually given to the background of the topic, and there's an in depth bibliography...the reviewer believes that for a few years to come back it will stay the classical presentation of summary harmonic analysis." Publicationes Mathematicae

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M ⊆ Ew∗m . Thus w has width m in E. Therefore w(E1 ) = E1,w Remark. It may have occurred to the reader that similar results for nilpotent groups could have been established without using generalized words at all. Is it worth all that trouble to make the small extension to virtually nilpotent groups? This is a moot point, but the techniques will be used in an essential way in later sections. 2. 20 Chapter 2. 2 Group ring stuﬀ Here we collect some module-theoretic results needed in the following section.

3. 1. Let E be a ﬁnitely generated group and A1 an abelian normal subgroup with E/A1 virtually nilpotent. 2 there exists n such that wE (A1 ) ⊆ Ew∗n . Thus to show that w has ﬁnite width in E it will suﬃce to show that it has ﬁnite width in E/wE (A1 ). e. that A1 ≤ w ∗ (E). 3, there exist normal subgroups A ≤ H of E, with A ≤ A1 , such that E/H is ﬁnite, A = γc (H) and A ∩ Z(H) = 1. Thus G = H/A is ﬁnitely generated nilpotent and A is ﬁnitely generated as a G-module. 4, and let K0 denote the subset consisting of E-invariant members of K.

Then p F ∩ F ≤ (F )p (exercise! ). As F is free abelian of rank d(d − 1)/2 it follows that G is elementary abelian of the same rank, while G/Z(G) is elementary abelian of rank d. 2. Commutators in p-groups 45 values in G (note that [x, y n ] = [xn , y] and [y, x] = [x, y −1 ] in G); while |γ2 (G)| = pd(d−1)/2 . Hence 2(2d − 1) 4 h dimG (Gγ 2 ) < < . d(d − 1) d−1 The preceding lemma now gives the result. A sharper result can be obtained by linear algebra, without using ﬁniteness. The following exercise implies a lower bound of [d/2] for the width of γ2 in Gd,p , and also in some torsion-free nilpotent groups, such as Fd /γ3 (Fd ) and its Mal’cev completion.