This is an important paper, establishing the existence of a monothetic (that is, topologically singly generated) topological group \(G\) such that every continuous action of \(G\) on a compact space has a fixed point. This is done by using the technique of concentration of measure on high-dimensional structures (see the book [M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Prog. Math. 152, Birkhäuser Verlag (1999)] for the most recently written and very fresh introduction to the topic). According to an important result by W. A. Veech [Bull. Am. Math. Soc. 83, 775-830 (1977; Zbl 0384.28018)], every locally compact group \(G\) acts freely on a suitable compact space. An opposite sort of behaviour is captured by saying that a topological group \(G\) is extremely amenable [E. Granirer, Math. Scand. 20, 93-113 (1967; Zbl 0204.03501)] or, as suggested in the paper under review, has the fixed point on compacta property (f.p.c.), if every continuous action of \(G\) on a compact space has a fixed point. Since such groups cannot be locally compact in view of Veech’s theorem, first examples were difficult to come by, and they indeed looked more like elaborately arranged pathologies, see [W. Herer and J. P. R. Christensen, Math. Ann. 213, 203-210 (1975; Zbl 0311.28002)] and also [W. Banaszczyk, Math. Ann. 264, 485-493 (1983; Zbl 0502.22010)]. However, recent developments have revealed that among the ‘massive’, infinite-dimensional, groups, the fixed point on compacta property is rather common. The technique of concentration of measure on high-dimensional structures had led M. Gromov and V. D. Milman [Am. J. Math. 105, 843-854 (1983; Zbl 0522.53039)] to establish the f.p.c. property of the unitary group \(U(l_2)_s\) of an infinite-dimensional Hilbert space with the strong operator topology, though here one needs to mention that, strictly speaking, these two authors had assumed an unnecessary condition of equicontinuity of actions and only the paper under review (Theorem 1.2) allows one to state the aforementioned results in its present generality. The reviewer had deduced from Ramsey theory the f.p.c. property for the orientation-preserving homeomorphism groups \(\text{Homeo}_+({\mathbb I})\) and \(\text{Homeo}_+({\mathbb R})\) [Trans. Am. Math. Soc. 350, No. 10, 4149-4165 (1998; Zbl 0911.54034)]. It was, in fact, the preprint version of the latter paper distributed in August 1996 and referred to in the paper that had prompted the appearance of the paper under review: it contained a question on the existence of a monothetic group with the f.p.c. property (removed in the final journal version). At the time, the reviewer was unaware of Banaszczyk’s examples [loc. cit.], as well as of their — easily proved – monotheticity. The author of the paper under review stresses that the result was known to him for a while, as well as (independently) to Furstenberg and Benjamin Weiss. In any case, it is perhaps the technique used for constructing the example that truly matters, rather than the existence of a group with the requested combination of properties!
In order to prove the result, the author refines the concept of a Lévy group introduced by Gromov and Milman [loc. cit.] in order to harness the phenomenon of concentration of measure on high-dimensional structures for dynamical purposes. A topological group is Lévy if it contains an everywhere dense union of an increasing sequence of compact subgroups \(G_i\), \(i\in{\mathbb N}\) with the following property: whenever \(A_i\subseteq G_i\) are Borel subsets the normalized Haar measures of which satisfy \(\limsup\mu_i(A_i)>0\), and whenever \(V\) is a neighbourhood of identity in \(G\), one has \(\lim\mu_i(V\cdot A_i\cap G_i)=1\). A theorem by Gromov and Milman in the form given to it by the author says that every Lévy group has the fixed point on compacta property. It remains to notice that the group \(L_1(X,U(1))\) formed by all measurable maps from a Lebesgue measure space \(X\) to the circle group \(U(1)\) and equipped with the obvious pointwise group operations and the \(L_1\)-metric is Lévy, the approximating compact subgroups being tori consisting of all step functions on a refining sequence of measurable partitions of \(X\). It is easy to see, using Kronecker type argument, that the above group is monothetic.
The following question was apparently first asked in print by Veech in [W. A. Veech, Am. J. Math. 90, 723-732 (1968; Zbl 0177.51204)] but obviously going back to the by now classical work of Følner and also Cotlar and Ricabarra: is it true that for every left syndetic subset \(S\) of the integers (that is, finitely many translations of \(S\) cover all of \(\mathbb Z\)) the set \(S-S\) is a Bohr neighbourhood of zero? While the question remains open, it is observed in the paper under review that the existence of a counterexample would follow if one had established the existence of a monothetic topological group that is minimally almost periodic and yet does not have the fixed point on compacta property.
Reviewer’s remark: for some later developments linking concentration of measure to the existence of fixed points in topological dynamics, see more recent papers by the reviewer [in: ‘Topological Dynamics and its Applications’. A Volume in Honor of Robert Ellis, Contemp. Math. 215, 83-99 (1998; Zbl 0891.54018); Amenable groups and measure concentration on spheres, C. R. Acad. Sci., Paris, Sér. I, 328, 669-674 (1999); Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space, Geometric and Funct. Anal., to appear, e-print available at http://xxx.lanl.gov/abs/math.FA/9903085].