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Extreme amenability of abelian \(L_0\) groups. (English) Zbl 1311.28002

The definition of amenability (introduced by J. von Neumann [Fundam. Math. 13, 73–116 (1929; JFM 55.0151.01); ibid. 13, 333 (1929; JFM 55.0151.02)]) was in terms of fixed point properties on compacta (f.p.c). In particular, a topological group \(G\) is defined to be extremely amenable in that every continuous action of \(G\) on a compact (Hausdorff) space \(X\) has a fixed point in \(X\). Discrete and locally compact groups do not generally have this property as they can be free-acting. Previously, the notion of extreme amenability theory had been well-defined only for semigroups, cf. T. Mitchell [Trans. Am. Math. Soc. 122, 195–202 (1966; Zbl 0146.12101)]. Classical amenability of groups concerns fixed points for continuous ‘affine’ actions of a group on a compact convex locally convex vector space.
V. Pestov had originally assumed in his definition of extreme amenability that \(G\) was already an amenable group. This restriction was subsequently avoided by introducing submeasures; these are are subadditive measures i.e., not having the property \(\mu(U)\cup \mu (V) = \mu(U) + \phi(V)\) for disjoint \(U\) and \(V\). Let \(\phi\) a submeasure on \((X, \mathcal{B}\)) where \(\phi : \mathcal{B} \to [0, +\infty)\) and \(\mathcal{B}\) a Boolean algebra of subsets of \(X\). The submeasure is called diffuse if for every \(\varepsilon > 0\) there is a finite cover of \(X\) by sets \(X_{i}\) each of which has submeasure less than \(\varepsilon\). D. Maharam [Ann. Math. (2) 48, 154–167 (1947; Zbl 0029.20401)] dealing with a ‘control problem’ characterised measure algebras (in the class of complete Boolean algebras), based on von Neumann [Problem 163, The Scottish Book (1937)]. She showed that a complete Boolean algebra is metrisable if and only if it supports a strictly positive submeasure with a ‘continuity’ property involving vanishing of infinite intersections.
The first example of an extreme amenable group was by W. Herer and J. P. R. Christensen [Math. Ann. 213, 203–210 (1975; Zbl 0311.28002)] using spectral theory, for the additive group \(\mathbb{R}\) with Lebesgue measure. A measure was called pathological (in that there is no finitely additive measure dominated by the submeasure cf., Maharam [loc. cit.]. They called a metrisable group exotic if it does not have any non trivial strongly continuous unitary representations in \(\mathcal{B}(H)\), \(H\) a Hilbert space and in this case proved the existence of a fixed point. Later other methods of constructing extremely amenable groups became available.
\(L^{0}(G,\phi)\) (\(X\) arbitrary compact Hausdorff and hidden) is defined to be the completion, for a a topology defined by convergence in submeasure, of the set of all submeasurable functions from \(X\) to \(G\) with finite range and it is given a group strucure by pointwise multiplication. In effect it is made up by submeasurable sections of a fibre-bundle with base \(X\). the fibres being copies of \(G\). \(L^{0}(\mathbb{Z},\mu)\), where \(\mu\) is the Lebesque counting measure, is not extremely amenable since \(\mathbb{Z}\) acts freely on the sections.
Extreme amenability theory is an extended version of the ‘H. Furstenberg program’ which concerned large Polish topological groups; cf. M. Gromov and V. D. Milman [Am. J. Math. 105, 843–854 (1983; Zbl 0522.53039)] who were concerned with sequences ofcompact subgroups with dense union. Existence of a concentration of measures was established leading to what are called Lévy groups (after P. Lévy’s [Leçons d’analyse fonctionnelle. Paris: Gauthier-Villars (1922; JFM 48.0453.01)] on concentration of measures on the surface of spheres in high-dimensional Euclidean spaces).
Theorems on extreme amenability of groups can sometimes be understood as Ramsey-type theorems cf. F. P. Ramsey [On a Problem of Formal Logic, Proceedings London Mathematical Society (1930)]. These can be something like ‘for every partition of a large structured object into classes one of the classes contains a large structured sub-object’, However, Ramsey gives no information about which class this would be. Ramsey theory here, for example, concerns finding monochromatic sets, cf. I. Farah and S. Solecki [J. Funct. Anal. 255, No. 2, 471–493 (2008; Zbl 1172.22002)] used a simplicial complex to to determine monochromatic sets for the Ramsey theory.
To construct simplicial complices, denoted \(K\), one considers a simplex as set comprising k points, taken to be vertices; a simplex having \(k+1\) vertices it is said to have dimension \(k\). It has a purely combinatorial description. It could also be closed under the operation of taking subsets. Denote the largest dimension of the component simplices by \(d\). Every d-dimensional simplicial complex can be realised as polyhedra in a \(\mathbb{R}^{2d+1}\); this is called the geometrical realisaton \(\|K\|\) of the simplicial complex. (Hence algebraic topology in that one maps simplical complices to a topological space.)
A (vertex) colouring of a graph \(\Gamma\) is or of a simplicial complex a function defined on the set of its vertices such that no two vertices connected with an edge are assigned the same value. The chromatic number \(\chi(\Gamma)\) is the minimal number of colors required to colour the vertices.
J. Lovácz [J. Combinatorial Theory Ser A 25 (1978)] proved the conjecture [M. Kneser, Jahresber. Deutsch. Math.-Verein 58 (1955)] that the graph – with all k-element subsets of \([n] = (1,2, \dots,n)\) as vertices and all pairs of disjoint sets as edges – has chromatic number \(n - 2k+2\). His proof used the Borsuk-Ulam theorem and has been adapted to obtain lower bounds for the chromatic number of finite graphs [K. Borsuk, Fundam. Math. 21, 35–38 (1933; JFM 59.1254.01)]. states that every continuous transformation of the sphere \(S^{n}\) into \(\mathbb{R}^{n}\) collapses a pair of antipodal points onto each other. The theory was initiated by L. A. Lyusternik [Monatsh. Math. Phys. 37, 125–130 (1930; JFM 56.1133.04)] follwed by L. G. Schnirel’mann and Borsuk in the 30’s. Schnirel’mann’s proof was in terms of a three-colouring of \(S^{2}\).
A barycentric subdivision \(sd(\mathcal{F})\) of a (partially ordered) simplicial complex \(\mathcal{F}\) is effected by declaring its vertices to be its set of nonempty simplices (while remaining ordered). Repeating the process, given a sufficient amount of subdivisions one may approximate any continuous function between polyhedra by simplicial maps.
K. Borsuk[Fundam. Math. 35, 217–234 (1948; Zbl 0032.12303)] dealt with finite embeddings of a simplicial complex in a finite dimensional compact metric space \(X\) of closed subsets on a simplicial complex such that the simplices coincide with those on the the \(X\). They both relate to thee same abstract simplicial complex.
Generalisations of Borsuk/Ulam to mappings \(S^{n} \to {\mathbb R}^{m}\) collapse a collection of points under the action a continuous function to a single point. Farah/Solecki [loc. cit.] use a version of Borsuk/Ulam due to A. Yu. Volovikov [Mat. Sb. (1979)] (see also M. Nakaoka [Osaka J. Math. 7, 443–449 (1970; Zbl 0218.57010)] and H. J. Munkholm [Math. Scand. 24, 167–185 (1969; Zbl 0186.57501)]); this relies on \(X\) being connected paracompact Hausdorff and a fixed-point-free mapping \(f:X \to X\) such that the cohomology indices \(H^{i}(X,\mathbb{Z}/p)\) all vanish mod \(p\). Farah/Solecki adapted Kneser graphs to a simplicial complex \(K\) with a grid of vertices embedded in the product of copies of \(\mathbb{Z}\) to get, for every continuous map \(f :\|Kp\| \to \mathbb{R}^{2d+1}\), a \(\mathbb{Z}/p\)-orbit that collapses to a single point in \(\|K\|\). They do this with a single prime \(p\) then go by induction to prove the general case using several primes \(p_{k}\) to get a simplicial complex \(\mathbb{Z}/p^{*n}\) which is the join of \(n\) copies. The vertices are denoted by \((i,r): i < n,r \in \mathbb{Z}/p\). The simplices are formed from subsets which form chains with respect to inclusion. The p-complex has \(s \in \mathbb{Z}/p\) acting as \( (i, r)\mapsto (i, s + r)\).
The author’s main theorem deals with supposedly general abelian groups \(G\); the reviewer asserts that for the article under review these have to be restricted to algebraically finitely generated \(G\). He tries to construct a comprehensive infinite system of undirected measurable graphs embedded in \(X\). He subdivides \(X\) into partitions \(\mathcal{P}\) of \(X\) into disjoint sets. The inductive process from \(G\) to \(L^{0}(G)\) extends actions \(\mathbb{Z} \to G\) to mappings \(\mathbb{Z}^{\mathcal P} \to L^{0}(G)\). His Definition 4 connects by edges the vertices of undirected graphs \((k_A: A\in \mathcal{P})\) and \((\ell_A: A \in \mathcal{P})\) if \(k_{A} = l_{A}+1\) on all of the space except a set of \(\phi\)-size less than \(\epsilon\). He uses \(\epsilon \in (0,1)\) as a parameter for fineness of the systems of graphs.
However his definition is improper as such; he needs to consider a symmetrisation of the graphs which then also provides edge-transitivity. The reviewer notes that the system connects vertices by edges if and only they lie within the same partition element for a given \(\epsilon\). It seems that his theorem is proved only \(\phi\)-almost everywhere on \(X\) and that he uses ultrafilter limits to select the graph he desires; much is dependent on the universality of his grid. His the basic argument seems to be valid but with an overcomplicated description.
The author’s main theorem is that \(L^{0}(G,\phi)\) is extremely amenable for ‘every’ abelian \(G\) and diffuse \(\phi\).
A subset \(A\) of an (additive) group \(H\) is called left syndetic if there is a compact \(K \subset H\) such that \(H = K+A\). In order to prove his main theorem the author, in his Lemma 5, uses a coloured version of Theorem 3.4.9 of [V. Pestov, Dynamics of infinite-dimensional groups. University Lecture Series 40. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1123.37003)]. Pestov gave necessary and sufficient conditions for extreme amenability of a topological group \(H\) (acting on \(X\)) in terms of left syndectic sets of \(H\), for example, if and only if for every left syndectic set \(S\subset H\), the set \(S - S\) is everywhere dense in \(H\).
To verify the second part of Lemma 5, {viz.} assuming that \(L^{0}(G,\phi)\) is not extremely amenable to deduce that the chromatic number of his graphs must be finite (except for partitions of small \(\phi\)-size). If \(G\) is an ‘arbitrary’ (i.e., finitely generated) abelian group then the \(L^{0}\) group with diffuse submeasure \(\phi\) has no finite upper bound on the chromatic numbers as \(\mathcal{P}\) runs over all the measurable partitions with \(\epsilon \in (0,1)\). To prove his assertion one would test whether a symmetric neighbourhood of \(S\) does not intersect \(gS\) when \(g \neq e\). This would show that the result is true \(\phi\)-almost everywhere in \(X\) (something like almost surely true).
The author uses the hypothesis that \(G\) is an abelian topological group with diffuse submeasure such that \(L^{0}(G)\) is not extremely amenable viz., there exists a syndectic set \(S\) and an element of \(L^{0}\) which is not in \(\overline{S-S}\). He assumes, without giving any justification, that there are a finite number of left translates by \(g_{i} \in G\) needed to cover \(G\) and concludes by induction that there are at most a finite number colours for his partition scheme for \(L^0\). The reviewer perceives the result is correct though his notation is extremely complicated. The reviewer prefers using the fibre structure of \(L^0(G)\) so that the elements of \(L^0(G)\) are measurable sections of the bundle.
To complete the proof his main theorem the author reverts to a Farah/Solecki simplicial complex. He proves that chromatic numbers of his graphs diverge when \(\phi\) is diffuse; he uses a Farah/Solecki partition \(\Gamma_n^{\mathcal P_n}(\phi)\) where \(\mathcal P_n\) is a partition of \(X\) into say \(k_n\) disjoint sets of submeasure less than \(1/n\). Given a diffused submeasure \(\phi\) and \(\epsilon > 0\) the author constructs a function \(F : \mathbb{N} \to \mathbb{N}\) which diverges slowly as \(n \to \infty\) and the chromatic numbers of the graphs are greater than \(F(n^{3 \over 2)})\). This \(F(n)\) is the analogue of inverse of the \(M_n\) as described in Farah/Solecki Lemma 4.3 (used when they showed that for a certain submeasure \(L^{0}\) is not Lévy).
It is not clear in the statement first part of Lemma 5, whether the author means ‘every’ or ‘any’ abelian group. He is trying to prove that if \(G\) is abelian and admits no finitely bounded colourings then it is not extremely amenable. He makes two different assumptions for his argument by contradiction, viz., that the graphs admit a finite colouring and also assumes without proof that \(L^{0}(\mathbb{Z}, \phi)\) is not extremely amenable (which should be the desired conclusion). This non-extreme amenability has not even been proved by Farah/Solecki even for compact groups (see their Proposition 4.5). Corollary 2 is suspect as stated.

MSC:

28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
28A60 Measures on Boolean rings, measure algebras
05C55 Generalized Ramsey theory
43A07 Means on groups, semigroups, etc.; amenable groups
54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.)
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