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.


03E15 Descriptive set theory
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
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[1] Abért, Symmetric groups as products of abelian subgroups, Bull. London Math. Soc. 34 pp 451– (2002) · Zbl 1035.20001 · doi:10.1112/S0024609302001042
[2] Akin, Generically there is but one self homeomorphism of the Cantor set (2006)
[3] Akin, Dynamics of topologically generic homeomorphisms, Mem. Amer. Math. Soc. 164 (2003)
[4] Anderson, The algebraic simplicity of certain groups of homeomorphisms, Amer. J. Math. 80 pp 955– (1958) · Zbl 0090.38802 · doi:10.2307/2372842
[5] Becker, The descriptive set theory of Polish group actions (1996) · Zbl 0949.54052 · doi:10.1017/CBO9780511735264
[6] Bekka, Kazhdan’s property (T) (2003)
[7] Bergman, Generating infinite symmetric groups, Bull. London Math. Soc. 38 pp 429– (2006) · Zbl 1103.20003 · doi:10.1112/S0024609305018308
[8] Bhattacharjee, A locally finite dense subgroup acting on the random graph, Forum Math. 17 pp 513– (2005) · Zbl 1093.20002 · doi:10.1515/form.2005.17.3.513
[9] Bryant, The small index property for free groups and relatively free groups, J. London Math. Soc. 55 ((2)) pp 363– (1997) · Zbl 0867.20032 · doi:10.1112/S0024610796004711
[10] Choksi, Baire category in spaces of measures, unitary operators and transformations, in: Invariant subspaces and allied topics pp 147– (1990)
[11] de Cornulier, Strongly bounded groups and infinite powers of finite groups, Comm. Algebra. · Zbl 1125.20023
[12] Droste, Uncountable cofinalities of permutation groups, J. London Math. Soc. 71 ((2)) pp 335– (2005) · Zbl 1070.20001 · doi:10.1112/S0024610704006167
[13] Droste, Generating automorphism groups of chains, Forum Math. 17 pp 699– (2005) · Zbl 1093.20016 · doi:10.1515/form.2005.17.4.699
[14] Dudley, Continuity of homomorphisms, Duke Math. J. 28 pp 587– (1961) · Zbl 0103.01702 · doi:10.1215/S0012-7094-61-02859-9
[15] Foreman, An anti-classification theorem for ergodic measure preserving transformations, J. Eur. Math. Soc. (JEMS) 6 pp 277– (2004) · Zbl 1063.37004 · doi:10.4171/JEMS/10
[16] Fräissé, Theory of relations (1986)
[17] Gartside, Ubiquity of free subgroups, Bull. London Math. Soc 35 pp 624– (2003) · Zbl 1045.22021 · doi:10.1112/S0024609303002194
[18] Gaughan, Topological group structures of infinite symmetric groups, Proc. Natl. Acad. Sci. USA 58 pp 907– (1967) · Zbl 0153.04301 · doi:10.1073/pnas.58.3.907
[19] Glasner, The topological Rohlin property and topological entropy, Amer. J. Math. 123 pp 1055– (2000) · Zbl 1012.54042 · doi:10.1353/ajm.2001.0039
[20] Grzaslewicz, Density theorems for measurable transformations, Colloq. Math. 48 pp 245– (1984)
[21] Guran, On topological groups close to being Lindelöf, Soviet Math. Dokl. 23 pp 173– (1981) · Zbl 0478.22002
[22] Halmos, Lectures on ergodic theory (1956)
[23] Herwig, Extending partial automorphisms and the profinite topology on free groups, Trans. Amer. Math. Soc. 352 pp 1985– (2000) · Zbl 0947.20018 · doi:10.1090/S0002-9947-99-02374-0
[24] Hjorth, Classification and orbit equivalence relations (2000) · Zbl 0942.03056
[25] Hodges, Model theory (1993) · doi:10.1017/CBO9780511551574
[26] Hodges, The small index property for {\(\omega\)}-stable {\(\omega\)}-categorical structures and for the random graph, J. London Math. Soc. 48 ((2)) pp 204– (1993) · Zbl 0788.03039 · doi:10.1112/jlms/s2-48.2.204
[27] Hofmann, The structure of compact groups. A primer for the student - a handbook for the expert (1998) · Zbl 0919.22001
[28] Hrushovski, Extending partial isomorphisms of graphs, Combinatorica 12 pp 411– (1992) · Zbl 0767.05053 · doi:10.1007/BF01305233
[29] Ivanov, Generic expansions of {\(\omega\)}-categorical structures and semantics of generalized quantifies, J. Symbolic Logic 64 pp 775– (1999) · Zbl 0930.03034 · doi:10.2307/2586500
[30] Ivanov, Strongly bounded automorphism groups, Colloq. Math. · Zbl 1098.20003
[31] Kechris, Classical descriptive set theory (1995) · doi:10.1007/978-1-4612-4190-4
[32] Kechris, Actions of Polish groups and classification problems, in: Analysis and logic (2002) · Zbl 1022.22003
[33] Kechris, Fräi ssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal. 15 pp 106– (2005) · Zbl 1084.54014 · doi:10.1007/s00039-005-0503-1
[34] Kechris, A strong generic ergodicity property of unitary and self-adjoint operators, Ergod. Theory Dynam. Systems 21 pp 1459– (2001) · Zbl 1062.47514 · doi:10.1017/S0143385701001705
[35] Kuske, Generic automorphisms of the universal partial order, Proc. Amer. Math. Soc. 129 pp 1939– (2001) · Zbl 0983.06004 · doi:10.1090/S0002-9939-00-05778-6
[36] Lascar, Les beaux automorphismes, Arch. Math. Logic 31 pp 55– (1991) · Zbl 0766.03022 · doi:10.1007/BF01370694
[37] Macpherson, Groups of automorphisms of 0-categorical structures, Quart.J. Math. Oxford 37 pp 449– (1986) · Zbl 0611.03014 · doi:10.1093/qmath/37.4.449
[38] Macpherson, Comeagre conjugacy classes and free products with amalgamation, Discrete Math. 291 pp 135– (2005) · Zbl 1058.03035 · doi:10.1016/j.disc.2004.04.025
[39] Möller, The automorphism groups of regular trees, J. London Math. Soc. 43 ((2)) pp 236– (1991) · Zbl 0681.20004 · doi:10.1112/jlms/s2-43.2.236
[40] Neumann, Groups coverable by permutable subsets, J. London Math. Soc. 29 pp 236– (1954) · Zbl 0055.01604 · doi:10.1112/jlms/s1-29.2.236
[41] Prasad, Generating dense subgroups of measure preserving transformations, Proc. Amer. Math. Soc. 83 pp 286– (1981) · Zbl 0468.28019 · doi:10.1090/S0002-9939-1981-0624915-2
[42] Rosendal, Automatic continuity of group homomorphisms and discrete groups with the fixed point on metric compacta property, Israel J. Math.
[43] Schmerl, Generic automorphisms and graph coloring, Discrete Math. 291 pp 235– (2005) · Zbl 1058.03036 · doi:10.1016/j.disc.2004.04.030
[44] Serre, Trees (2003) · Zbl 1013.20001
[45] Solecki, Extending partial isometries, Israel J. Math. · Zbl 1124.54012
[46] Tkachenko, Topological groups: between compactness and š 0-boundedness, in: Recent progress in general topology, II pp 515– (2002)
[47] Truss, Generic automorphisms of homogeneous structures, Proc. London Math. Soc. 65 ((3)) pp 121– (1992) · Zbl 0723.20001 · doi:10.1112/plms/s3-65.1.121
[48] Vershik, Random metric spaces and universality, Uspekhi Mat. Nauk 59 pp 65– (2004)
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