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Automorphism Groups of Countably Categorical Linear Orders are Extremely Amenable

François Gilbert Dorais, Steven Gubkin, Daniel McDonald, Manuel Rivera
2012 Order  
We show that the automorphism groups of countably categorical linear orders are extremely amenable.  ...  Using methods of Kechris, Pestov, and Todorcevic, we use this fact to derive a structural Ramsey theorem for certain families of finite ordered structures with finitely many partial equivalence relations  ...  In this paper, we generalize this result to the class of all countably categorical linear orders. Theorem 1.1. The automorphism group of a countably categorical linear order is extremely amenable.  ... 
doi:10.1007/s11083-012-9252-6 fatcat:tuih3e4fjvau5mnhgqvwqhmk3i

New Ramsey Classes from Old [article]

Manuel Bodirsky
2012 arXiv   pre-print
We prove that when C_1 and C_2 are Ramsey, then C_1 wedge C_2 is also Ramsey.  ...  We also discuss variations of this statement, and give several examples of new Ramsey classes derived from those general results.  ...  Many thanks to François Bossière, Jan Foniok, András Pongrácz, and Miodrag Sokić for their valuable comments on earlier versions of this text.  ... 
arXiv:1204.3258v2 fatcat:jt2qx6x76vc7znkrxptgug2oli

Automorphism groups and Ramsey properties of sparse graphs

David M. Evans, Jan Hubička, Jaroslav Nešetřil
2019 Proceedings of the London Mathematical Society  
Resolving one of the open questions in the area, we show that Hrushovski's example of an ω-categorical sparse graph has no ω-categorical expansion with extremely amenable automorphism group.  ...  We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory.  ...  For example, it asks, given a countable ω-categorical structure M , does there exist an ωcategorical expansion of M whose automorphism group is extremely amenable?  ... 
doi:10.1112/plms.12238 fatcat:trv2g76rnrbbjjnm5ly5oj7rxa

Topological dynamics of automorphism group of countably categorical structures [article]

Aleksander Ivanov
2014 arXiv   pre-print
We consider automorphism groups of some countably categorical structures and their precompact expansions.  ...  We prove that automorphism groups of omega-stable omega-categorical structures have metrizable universal minimal flows. We also study amenability of these groups.  ...  Since S ∞ and the automorphism group of an ω-dimensional vector space over a finite field are amenable ([2]), the group of automorphisms of M 0 ∪ D induced by Aut(M) is amenable too.  ... 
arXiv:1412.6657v1 fatcat:dolv2cwy5vfzddojl2d6tzogtq

Canonical Functions: a proof via topological dynamics [article]

Manuel Bodirsky, Michael Pinsker
2020 arXiv   pre-print
Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties.  ...  We moreover present equivalent algebraic characterisations of canonicity.  ...  action of its automorphism group on tuples, and a countable structure is ω-categorical if and only if its automorphism group is oligomorphic.  ... 
arXiv:1610.09660v3 fatcat:wsly2lcjcbfkzozkovyvn23vau

Nice enumerations and extreme amenability [article]

Aleksander Ivanov
2014 arXiv   pre-print
We study properties related to nice enumerability of countably categorical structures and properties related to extreme amenability of automorphism groups of these structures.  ...  In particular the example of Section 3 is transported to arXiv: 1403.7610. This version is not final.  ...  Let (L, <) be a countable linear order on a subset of an ω-categorical structure M.  ... 
arXiv:1310.7116v2 fatcat:yhrv7xzoonhsthxo3xaxi5dn44

On first order amenability [article]

Ehud Hrushovski, Krzysztof Krupiński, Anand Pillay
2021 arXiv   pre-print
Amenability of T follows from amenability of the (topological) group Aut(M) for all sufficiently large ℵ_0-homogeneous countable models M of T (assuming T to be countable), but is radically less restrictive  ...  We introduce the notion of first order amenability, as a property of a first order theory T: every complete type over ∅, in possibly infinitely many variables, extends to an automorphism-invariant global  ...  Let us comment on the relation between extreme amenability of the automorphism group of an ω-categorical, countable structure M as considered in [10] (which we call KPT-extreme amenability) and extreme  ... 
arXiv:2004.08306v2 fatcat:sxtdiheljndgrdsdiqyt36rwgm

A survey on structural Ramsey theory and topological dynamics with the Kechris-Pestov-Todorcevic correspondence in mind [article]

Lionel Nguyen Van Thé
2014 arXiv   pre-print
The purpose of the present paper is to present a self-contained survey of the corresponding developments.  ...  The following are equivalent: i) The group G is extremely amenable. ii) The class Age(F) has the Ramsey property and consists of rigid elements.  ...  A direct application of those results allowed to find a wealth of extremely amenable groups and of universal minimal flows.  ... 
arXiv:1412.3254v2 fatcat:nlaso4d63zdipiltn4hqbii77i

Ramsey Classes: Examples and Constructions [article]

Manuel Bodirsky
2015 arXiv   pre-print
Ramsey classes have recently attracted attention due to a surprising link with the notion of extreme amenability from topological dynamics.  ...  This article is concerned with classes of relational structures that are closed under taking substructures and isomorphism, that have the joint embedding property, and that furthermore have the Ramsey  ...  I also want to thank Antoine Mottet and András Pongrácz for discussions around the topic of this survey.  ... 
arXiv:1502.05146v3 fatcat:uvyaqmd5jrccvorsx3s36kmzei

A survey of homogeneous structures

Dugald Macpherson
2011 Discrete Mathematics  
A relational first order structure is homogeneous if it is countable (possibly finite) and every isomorphism between finite substructures extends to an automorphism.  ...  In the particular case of (Q, <), the back-and-forth method is not needed: a piecewise linear automorphism extending f could be constructed directly.  ...  Some of the theory connecting bi-interpretability and the topology on the automorphism group (see Theorem 5.1.2) has analogues for these notions of definability, with the automorphism group replaced by  ... 
doi:10.1016/j.disc.2011.01.024 fatcat:fqpfnc25hjdajekkiyaeszw6ye

A Survey On Structural Ramsey Theory And Topological Dynamics With The Kechris-Pestov-Todorcevic Correspondence In Mind

Lionel Nguyen Van Thé
2015 Zenodo  
The purpose of the present paper is to present a self-contained survey of the corresponding developments.  ...  A direct application of those results allowed to find a wealth of extremely amenable groups and of universal minimal flows.  ...  Let us simply mention that, concerning extreme amenability, it took a long time before even proving that such groups exist, but that several nonlocally compact transformation groups are now known to be  ... 
doi:10.5281/zenodo.1471599 fatcat:hlaz3o7lkzf6dh3sjgxqvfhp7a

Automorphism groups of finite topological rank [article]

Itay Kaplan, Pierre Simon
2019 arXiv   pre-print
In fact, we are not aware of any ω-categorical structure to which they do not apply (assuming the automorphism group has no compact quotients). We end with a few questions and conjectures.  ...  We then show how finite topological rank of the automorphism group of an ω-categorical structure can go down to reducts.  ...  We would also like to thank the organizers of the 2016 Permutation Groups workshop in Banff, during which those interactions took place.  ... 
arXiv:1709.01918v5 fatcat:hduwxtxlbndo7pvsfc3xw3w5he

Glasner's problem for Polish groups with metrizable universal minimal flow [article]

Lionel Nguyen Van Thé
2018 arXiv   pre-print
A problem of Glasner, now known as Glasner's problem, asks whether every minimally almost periodic, monothetic, Polish groups is extremely amenable.  ...  The purpose of this short note is to observe that a positive answer is obtained under the additional assumption that the universal minimal flow is metrizable.  ...  I am grateful to Itaï Ben Yaacov for his reference to [BY16] and his explanations regarding it; to Julien Melleray for the references regarding simplicity of automorphism groups; to Todor Tsankov for  ... 
arXiv:1705.05739v2 fatcat:jwovg5wjubcahkptwx6tukucum

Automorphism Groups of Generic Structures: Extreme Amenability and Amenability [article]

Zaniar Ghadernezhad, Hamed Khalilian, Massoud Pourmahdian
2016 arXiv   pre-print
Using these correspondences, we prove that automorphism groups of ordered Hrushovski generic graphs are not extremely amenable in both cases of collapsed and uncollapsed.  ...  We investigate correspondences between extreme amenability and amenability of automorphism groups of Fra\"iss\'e-Hrushovski generic structures that are obtained from smooth classes, and their Ramsey type  ...  Extreme amenability of automorphism groups of generic structures In [7] , the general correspondence between the extreme amenability of the automorphism group of an ordered Fraïssé structure and the Ramsey  ... 
arXiv:1508.04628v2 fatcat:2kqqyoa6gvgg7j2uopiqlwuyre

Polish groups with metrizable universal minimal flows [article]

Julien Melleray, Lionel Nguyen Van Thé, Todor Tsankov
2018 arXiv   pre-print
We prove that if the universal minimal flow of a Polish group G is metrizable and contains a G_δ orbit G · x_0, then it is isomorphic to the completion of the homogeneous space G/G_x_0 and show how this  ...  We also investigate universal minimal proximal flows and describe concrete representations of them in a number of examples.  ...  group of the dense countable linear order (Q, <) is extremely amenable (i.e., every time it acts continuously on a compact space, there is a fixed point) and produced the first interesting example of  ... 
arXiv:1404.6167v3 fatcat:sj6vhgmsorbhlixzjrcapgoa44
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