Non-genericity phenomena in ordered Fraïssé classes

Slutsky, Konstantin

doi:10.2178/jsl/1344862171

Cited by 8 publications

(10 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Clearly, this is a generalization of the notion of the diagonal conjugacy class, and it is still not known whether there exists a Polish group G such that for some n ≥ 2 there is a non-meager ndimensional topological similarity class, but all n-dimensional diagonal conjugacy classes are meager. Generalizing methods and results of Slutsky [16], we show (Theorem 5.5) that if M is the Fraïssé limit of a Fraïssé class F that is a full order expansion and that satisfies certain additional conditions, then all 2-dimensional topological similarity classes in Aut(M) are meager. In particular, this is true if K is a class with free amalgamation, or the class of ordered tournaments (Theorem 5.7.…”

Section: Introductionmentioning

confidence: 75%

“…The two-dimensional case. Similarity classes Slutsky [16] showed that every 2-dimensional topological similarity class in Aut(É) is meager. In this section, we extract from Slutsky's arguments a general condition on a structure M that implies that every 2-dimensional topological similarity class in Aut(M) is meager (Theorem 5.5).…”

Section: Claim the Map Q Preserves Smentioning

confidence: 99%

“…Let F be a Fraïssé class. Generalizing the terminology introduced in [16], for A, B ∈ F with B ⊆ A, and a partial automorphism q of A such that def(q) ∩ B = ∅, we say that B is free from q if for every n, every relation symbol R in the signature of F of arity n, for all x 1 , . . .…”

Section: Claim the Map Q Preserves Smentioning

confidence: 99%

“…It seems that these are, except for Aut(É) (see Truss [17]), the only known order classes such that the automorphism group of the limit has a comeager conjugacy class. We also give (Theorem 3.8) a short and elementary proof of a result of Slutsky [16] who showed that the class (QU ≺ ) 1 of partial automorphisms of ordered metric spaces with rational distances has no WAP.…”

Section: Introductionmentioning

confidence: 98%

See 3 more Smart Citations

Ordered structures and large conjugacy classes

Kwiatkowska

Malicki

2020

Journal of Algebra

View full text Add to dashboard Cite

This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraïssé limit) that is extremely amenable, and has ample generics. As Fraïssé limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraïssé limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager 2-dimensional diagonal conjugacy class. We also provide general conditions implying that there is no comeager conjugacy class, comeager 2-dimensional diagonal conjugacy class or non-meager 2-dimensional topological similarity class in the automorphism group of an ordered Fraïssé limit. We provide a number of applications of these results.2010 Mathematics Subject Classification. 03E15, 54H11.

show abstract

Section: Introductionmentioning

confidence: 75%

Section: Claim the Map Q Preserves Smentioning

confidence: 99%

Section: Claim the Map Q Preserves Smentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

Ordered structures and large conjugacy classes

Kwiatkowska

Malicki

2020

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…An interesting example is Q, < which admits generics but not even 2-generics (Hodkinson, see [60] and [56]). …”

Section: B)mentioning

confidence: 99%

Dynamics of non-archimedean Polish groups

Kechris¹

European Congress of Mathematics Kraków, 2 – 7 July, 2012

View full text Add to dashboard Cite

Abstract.A topological group G is Polish if its topology admits a compatible separable complete metric. Such a group is non-archimedean if it has a basis at the identity that consists of open subgroups. This class of Polish groups includes the profinite groups and (Qp, +) but our main interest here will be on non-locally compact groups.In recent years there has been considerable activity in the study of the dynamics of Polish non-archimedean groups and this has led to interesting interactions between logic, finite combinatorics, group theory, topological dynamics, ergodic theory and representation theory. In this paper I will give a survey of some of the main directions in this area of research.

show abstract

Generic representations of abelian groups and extreme amenability

Melleray

Tsankov

2013

Isr. J. Math.

View full text Add to dashboard Cite

To cite this version:Julien Melleray, Todor Tsankov. Generic representations of abelian groups and extreme amenability.Israël Journal of Mathematics, The Hebrew University Magnes Press, 2013, 198 (1), http://link.springer.com/article/10.1007/s11856-013-0036-5. <10.1007/s11856-013-0036-5>. GENERIC REPRESENTATIONS OF ABELIAN GROUPS AND EXTREME AMENABILITY JULIEN MELLERAY AND TODOR TSANKOVABSTRACT. If G is a Polish group and Γ is a countable group, denote by Hom(Γ, G) the space of all homomorphisms Γ → G. We study properties of the group π(Γ) for the generic π ∈ Hom(Γ, G), when Γ is abelian and G is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on Γ, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic π(Γ); in the other two, we show that the generic π(Γ) is extremely amenable. We also show that if Γ is torsion-free, the centralizer of the generic π is as small as possible, extending a result of Chacon-Schwartzbauer from ergodic theory.

show abstract

Non-genericity phenomena in ordered Fraïssé classes

Abstract: ABSTRACT. We show that every two-dimensional class of topological similarity, and hence every diagonal conjugacy class of pairs, is meager in the group of order preserving bijections of the rationals and in the group of automorphisms of the ordered rational Urysohn space.

Cited by 8 publications

References 7 publications

Ordered structures and large conjugacy classes

Ordered structures and large conjugacy classes

Dynamics of non-archimedean Polish groups

Generic representations of abelian groups and extreme amenability

Contact Info

Product

Resources

About