Hilbert's Fifth Problem and Related Topics

Description

This book consists of two parts, the first much longer than the second. The first part, addressed in the bulk of this review, is a discussion of the structure theory of locally compact groups and related topics connected with "Hilbert's fifth problem''. The second part is a miscellaneous selection of topics, in which respect it is similar to several previous texts based on the author's weblog. The reader interested in group theory and/or additive combinatorics will find much of interest in this second part, but so will others. Whilst some of the material (the Jordan-Schur theorem, Ado's theorem, the Peter-Weyl theorem) is quite standard, other parts are far less so. For example, in Chapter 15 one may find an interesting example of a local group which is not globalisable, and in Chapter 19 one may find an ultrafilter argument employed to prove a quantitative version of the Nullstellensatz, in a form considered by Chang: If P1,…,Pr∈Q[X1,…,Xd]P_1,\dots, P_r \in \bold{Q}[X_1,\dots, X_d] and if the system P1(z)=⋯=Pr(z)=0P_1(z) = \dots = P_r(z) = 0 admits a solution z∈Cdz \in \Bbb{C}^d, then it also admits a solution z∈Q¯dz \in \overline{\bold{Q}}^d, which is furthermore bounded in height by a polynomial function of the heights of the coefficients of P1,…,PrP_1,\dots, P_r.

We turn now to the first part, which composes the majority of the book. This part resulted from a course given by the author at UCLA, the aim of which was to explain the theory surrounding Hilbert's fifth problem and its application to results in additive combinatorics, specifically the structure theorem for approximate groups ([E. Breuillard, B. Green and T. C. Tao, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 115–221; MR3090256]—the reader may consult this survey for more information about what an approximate group is; we will not say anything about it here). Much of the theory in question was laid out in the 1950s by Gleason, Yamabe, Montgomery, Zippin and others. Though respectable accounts have been written since, it has long retained an aura of mystery, even after M. Gromov [Inst. Hautes Études Sci. Publ. Math. No. 53 (1981), 53–73; MR0623534] provided a stunning application by using the theory to establish that all groups of polynomial growth are virtually nilpotent. This book provides an accessible account, specifically tailored to the recent applications in additive combinatorics, and as such is a very welcome addition to the literature.
As pointed out in the introduction, Hilbert's fifth problem does not refer to any single problem but rather to a collection of (at one time conjectural) statements asserting that some weak hypothesis on a group or group action implies far greater structure. At least three statements of this type are discussed in the book:
Theorem 1 (Gleason-Yamabe theorem). Let GG be a locally compact group. Then GG is "almost Lie'' in the following sense: there is an open subgroup G′G' of GG such that for any open neighbourhood of the identity U⊂G′U \subset G' there is a compact normal subgroup KK of G′G' in UU such that G′/KG'/K is isomorphic to a Lie group.
Theorem 2. Let GG be a locally compact σ\sigma-compact group acting transitively faithfully and continuously on a connected topological manifold XX. Then GG is isomorphic to a Lie group. (As an aside, we note that the Hilbert-Smith conjecture posits that the transitivity assumption can be dropped. This conjecture is discussed in Chapter 17 of the book; it was recently established in dimension 3 by J. Pardon [J. Amer. Math. Soc. 26 (2013), no. 3, 879–899; MR3037790]. As another aside, we remark that it was this statement that Gromov used in his paper cited above.)
Theorem 3 (Montgomery-Zippin, Yamabe). Suppose that GG is a locally Euclidean topological group. Then GG is isomorphic to a Lie group.
All three of these results are established in Part I of the book. The main effort, which occupies Chapters 2 through 5, consists of proving Theorem 1, which is the deepest statement; the other two results are deduced as consequences in Chapter 6. Chapters 7 and 8 are then devoted to a discussion of a wonderful principle discovered by E. Hrushovski [J. Amer. Math. Soc. 25 (2012), no. 1, 189–243; MR2833482]: an ultraproduct of approximate groups has a "Lie model''. Neither the proof nor the statement given here is quite the original one of Hrushovski, but are slightly more down-to-earth ones contained in [E. Breuillard, B. Green and T. C. Tao, op. cit.], relying on a lemma of T. Sanders [J. Aust. Math. Soc. 89 (2010), no. 1, 127–132; MR2727067] (obtained in a similar form by E. S. Croot III and O. Sisask [Geom. Funct. Anal. 20 (2010), no. 6, 1367–1396; MR2738997]). In Chapter 9, an exposition of many of the results from [E. Breuillard, B. Green and T. C. Tao, op. cit.] is given. Finally, in Chapter 10 some applications are given.
In summary, this book is a very useful and well-written contribution to the literature, greatly clarifying the whole area. It provides a fine introduction for anyone wishing to understand recent advances in additive combinatorics, written by the main player in the field, but it will also be of great help to those wishing to learn about the theory of locally compact groups for its own sake.

Reviewed by Ben Joseph Green

cf.

Elemer E Rosinger, What is the situation with Hilbert's Fifth Problem?(version: 2018-02-07).