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Turbulence, amalgamation, and generic automorphisms of homogeneous structures. (English) Zbl 1118.03042

The authors present a thorough, systematic and unified treatment of the problems whether the automorphism groups of certain countable structures have dense or, moreover, dense \(G_{\delta}\) conjugacy classes. Let \(\mathcal K\) be a class of finite structures in a fixed countable signature. We say that \(\mathcal K\) is a Fraïssé class if it has the following properties:
(i) \(\mathcal K\) is hereditary, that is, \(A \leq B \in \mathcal K\) implies \(A\in\mathcal K\) (where \(A\leq B\) means \(A\) can be embedded into \(B\));
(ii) \(\mathcal K\) satisfies the joint embedding property, that is, if \(A,B\in\mathcal K\) then there is a \(C\in\mathcal K\) with \(A, B\leq C\);
(iii) \(\mathcal K\) satisfies the amalgamation property, that is, if \(f\colon A \rightarrow B\) and \(g\colon A \rightarrow C\) are embeddings with \(A,B,C \in \mathcal K\) then there exists \(D\in\mathcal K\) and embeddings \(r\colon B \rightarrow D\) and \(s\colon C \rightarrow D\) with \(r\circ f=s\circ g\);
(iv) \(\mathcal K\) contains, up to isomorphism, only countably many structures, and contains structures of arbitrarily large (finite) cardinality.
For any Fraïssé class \(\mathcal K\) there is a corresponding Fraïssé limit K= Flim\((\mathcal K)\), which is the unique countably infinite structure such that:
(a) K is locally finite, that is, finitely generated substructures of K are finite;
(b) K is ultrahomogeneous, that is, any isomorphism between finite substructures extends to an automorphism of K;
(c) Age\((\)K\()=\mathcal K\), where Age\((\)K\()\) is the class of all finite structures that can be embedded in K.
A countably infinite structure K satisfying properties (a) and (b) is called a Fraïssé structure and the correspondence \({\mathcal K} \mapsto\) Flim\(({\mathcal K})\) and K \(\mapsto\) Age(K) is a canonical bijection between Fraïssé classes and Fraïssé structures. Examples of Fraïssé structures include the trivial structure \((\mathbb N,=)\), the random graph, the order type of the rational numbers, the countable atomless Boolean algebra, and the rational Urysohn space. The latter is the Fraïssé limit of the class of finite metric spaces with rational distances.
Let \(\mathcal{K}\) be a Fraïssé class and let K be the Fraïssé limit of \(\mathcal{K}\). Let \(\mathcal{K}_{p}\) denote the class of all systems \(\mathcal{S} = \left< {A}, \psi \colon {B} \rightarrow {C} \right>\) where \({A}, {B}, {C} \in \mathcal{K}\), \( {B}, {C} \subseteq {A}\) and \(\psi\) is an isomorphism of \( {B}\) and \( {C}\). An embedding of \(\mathcal{S}\) into \(\mathcal{T} = \left< {D}, \phi \colon {E} \rightarrow {F} \right> \in \mathcal{K}_{p}\) is defined as an embedding \(f \colon {A} \rightarrow {D}\) such that \(f\) embeds \( {B}\) into \( {E}\) and \( {C}\) into \( {F}\) and \(f \circ \psi \subseteq \phi \circ f\).
The first main result of the paper states that for a Fraïssé class \(\mathcal{K}\) with its Fraïssé limit K the following are equivalent: (i) there is a dense conjugacy class in Aut(K); (ii) \(\mathcal{K}_{p}\) satisfies the joint embedding property. Since \(\mathcal{K}_{p}\) has the joint embedding property if \(\mathcal{K}\) is the class of
(a) finite metric spaces with rational distances,
(b) finite Boolean algebras,
(c) finite measure Boolean algebras with rational measure,
as a corollary of this theorem one obtains that the following Polish groups have dense conjugacy classes:
(a’) the isometry group of the Urysohn space,
(b’) the homeomorphism group of the Cantor space (see also E. Glasner and B. Weiss [Am.J. Math. 123, No. 6, 1055–1070 (2001; Zbl 1012.54042)] and E.Akin, M.Hurley and J.A. Kennedy [Mem.Am.Math.Soc.783 (2003; Zbl 1022.37010)]),
(c’) the automorphism group of a standard measure space (this result is known as Rokhlin property).
In order to characterize those Fraïssé classes \(\mathcal{K}\) where Aut(K) has a dense \(G_{\delta}\) conjugacy class the authors formulate the notion weak amalgamation property for \(\mathcal{K}_{p}\). As the second main result, it is obtained that Aut(K) has a dense \(G_{\delta}\) conjugacy class if and only if \(\mathcal{K}_{p}\) has the joint embedding property and the weak amalgamation property.
As a corollary, the authors show in particular that the homeomorphism group of the Cantor space has a dense \(G_{\delta}\) conjugacy class.
The existence of ample generics is also discussed. A Polish group \(G\) is said to have ample generic elements if for each \(n<\omega\) there is a comeager orbit for the diagonal conjugacy action \(g \cdot (g_{1}, \dots, g_{n}) = (gg_{1}g^{-1}, \dots, gg_{n}g^{-1})\) of \(G\) on \(G^{n}\). The authors find two new groups with ample generics, which are automorphism groups of not \(\aleph_{0}\)-categorical structures: the group of Haar measure-preserving homeomorphisms of the Cantor space and the group of Lipschitz homeomorphisms of the Baire space. Several consequences of having ample generics are established: if \(G\) is a Polish group with ample generic elements then
(1) \(G\) has the small index property, i.e. any subgroup of \(G\) with index less than continuum is open;
(2) \(G\) is not the union of countably many non-open subgroups;
(3) every homomorphism of \(G\) into a topological group with uniform Souslin number at most continuum is necessarily continuous.
These results generalize some results of W. Hodges, I. Hodkinson, D. Lascar and S. Shelah [J. Lond. Math. Soc., II. Ser. 48, No. 2, 204–218 (1993; Zbl 0788.03039)].
Apart from the theorems recalled in the present review, the paper abounds with more sophisticated results connected to the existence of special dense conjugacy classes, to strengthened small index properties and to automatic continuity. The paper concludes with a list of related open problems.

MSC:

03E15 Descriptive set theory
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
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