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Universal minimal flows of automorphism groups

A.S. Kechris, V. Pestov, S. Todorcevic
<span title="">2003</span> <i title="National Library of Serbia"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wakhi7dxmneolkwa7dbwayp2ga" style="color: black;">Bulletin Classe des sciences mathematiques et natturalles</a> </i> &nbsp;
graph, the automorphism group of the ∞-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra,etc.  ...  As result we compute universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free  ...  (ii) (a) G preserves a linear ordering and (b) G has the Ramsey property. Theorem 3 . 3 The automorphism groups of the following structures are extremely amenable:(i) The random ordered graph.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2298/bmat0328093k">doi:10.2298/bmat0328093k</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vrzdswkkybhbrb3isu2khbi4xa">fatcat:vrzdswkkybhbrb3isu2khbi4xa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808234914/http://authors.library.caltech.edu/38690/1/Kechris_2003p93.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/97/e1/97e1539cf56aa5b498d41b1e5f21e921166f92de.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2298/bmat0328093k"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Martin-Lof randomness, invariant measures and countable homogeneous structurs [article]

Willem L. Fouche
<span title="2012-05-02">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We use ideas from topological dynamics (amenability), combinatorics (structural Ramsey theory) and model theory (Fraïssé limits) to study closed amenable subgroups G of the symmetric group S_∞ of a countable  ...  This leads to an identification of the generic elements, in the sense of being Martin-Löf random, of these flows with respect to the constructed invariant measures.  ...  Acknowledgements The research is based upon work supported by the National Research Foundation (NRF) of South Africa.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1205.0386v1">arXiv:1205.0386v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cvlnun5ppfcbzdzipteldkmvgi">fatcat:cvlnun5ppfcbzdzipteldkmvgi</a> </span>
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The classification problem for automorphisms of C*-algebras [article]

Martino Lupini
<span title="2015-01-13">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We present an overview of the recent developments in the study of the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory.  ...  In fact it suffices to define maps R, S as above from trees to countable linear orders. One can then just take the tree of finite descending chains of the linear order.  ...  In order to verify the second condition above, it is enough to argue that one can detect the existence of an infinite branch of T by looking at group-theoretic properties of G T .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1501.03022v1">arXiv:1501.03022v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/33zelbst4zet5gg2ytnphpzoty">fatcat:33zelbst4zet5gg2ytnphpzoty</a> </span>
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THE CLASSIFICATION PROBLEM FOR AUTOMORPHISMS OF C*-ALGEBRAS

MARTINO LUPINI
<span title="">2015</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nxnim3i24fhcvkvgruk2dej7yu" style="color: black;">Bulletin of Symbolic Logic</a> </i> &nbsp;
We present an overview of the recent developments in the study of the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory.  ...  In fact it suffices to define maps R, S as above from trees to countable linear orders. One can then just take the tree of finite descending chains of the linear order.  ...  In order to verify the second condition above, it is enough to argue that one can detect the existence of an infinite branch of T by looking at group-theoretic properties of G T .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/bsl.2015.37">doi:10.1017/bsl.2015.37</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zzhkzsktp5e4fai4ottfrz6q7y">fatcat:zzhkzsktp5e4fai4ottfrz6q7y</a> </span>
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A survey of homogeneous structures

Dugald Macpherson
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
The article discusses connections between these topics, with an emphasis on examples, and on special properties of an amalgamation class which yield important consequences for the automorphism group. language  ...  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  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2011.01.024">doi:10.1016/j.disc.2011.01.024</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fqpfnc25hjdajekkiyaeszw6ye">fatcat:fqpfnc25hjdajekkiyaeszw6ye</a> </span>
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Page 632 of Mathematical Reviews Vol. , Issue 91B [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
G is a Scott system) and L is a countable or @)-like dense linear order with first element but no last element, then there is a recursively saturated model M of Pr such that t(M) = G and m(M) = L.  ...  In particular, the notion of unimodularity is defined, and it is pointed out that this is the crucial property required for carrying out certain computations on a pseudoplane.  ... 
<span class="external-identifiers"> </span>
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Page 756 of Mathematical Reviews Vol. , Issue 2004b [page]

<span title="">2004</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
theoretic properties of the group of computable automorphisms of a countable dense linear order.  ...  In this article the properties of the group Aut(Q), the classi- 03 MATHEMATICAL LOGIC AND FOUNDATIONS 756 cal automorphism group of a countable dense linear order, are compared with those of Aut,.  ... 
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Groups and Monoids of Cellular Automata [chapter]

Ville Salo
<span title="">2015</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Both purely group-theoretic properties and issues of decidability are considered. interested in a strong, structural kind of reversibility where there is a concrete inverse time-evolution rule g : X →  ...  of a single function on X, we have a countable discrete monoid M acting on X by functions.  ...  If G is the automorphism group of a subshift, is it also the automorphism group of a countable subshift? Are automorphism groups of countable subshifts amenable? Can they contain a copy of F 2 ?  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-47221-7_3">doi:10.1007/978-3-662-47221-7_3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vxmmfn6zijdszcoro3hxotfsee">fatcat:vxmmfn6zijdszcoro3hxotfsee</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190429134726/https://hal.inria.fr/hal-01442480/file/338243_1_En_3_Chapter.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/4d/72/4d7219decea7814c43538e8062dc905416315c5c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-47221-7_3"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Oligomorphic Permutation Groups [chapter]

Peter J. Cameron
<span title="">2009</span> <i title="WORLD SCIENTIFIC"> Perspectives in Mathematical Sciences II </i> &nbsp;
We give a spread of examples, describe results on the growth rate of the counting functions, discuss a graded algebra associated with an oligomorphic group, and finally discuss group-theoretic properties  ...  A permutation group G (acting on a set Ω, usually infinite) is said to be oligomorphic if G has only finitely many orbits on Ω n (the set of n-tuples of elements of Ω).  ...  The symmetric group S, the group A of order-preserving permutations of Q, and the automorphism group of the random graph all have the strong small index property [15, 19, 8] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/9789814273657_0003">doi:10.1142/9789814273657_0003</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kxt2dwfkpvc45kfjwjkb2j6qma">fatcat:kxt2dwfkpvc45kfjwjkb2j6qma</a> </span>
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Polish groups and Baire category methods

Julien Melleray
<span title="">2016</span> <i title="Cellule MathDoc/CEDRAM"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gwclsy5eqjd63f7qaqz4pfeigu" style="color: black;">Confluentes Mathematici (CM)</a> </i> &nbsp;
One last remark before forging ahead: there are many ways to turn a nonarchimedean Polish group into the automorphism group of a countable first-order structure; some properties of the structure will be  ...  As examples, one can cite the automorphism group of the random graph, the automorphism group of a countable free group...  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5802/cml.28">doi:10.5802/cml.28</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gvwwc5ddrzdpho33vpor23mxbm">fatcat:gvwwc5ddrzdpho33vpor23mxbm</a> </span>
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A survey on structural Ramsey theory and topological dynamics with the Kechris-Pestov-Todorcevic correspondence in mind [article]

Lionel Nguyen Van Thé
<span title="2014-12-12">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
As an immediate consequence, it triggered a new interest for structural Ramsey theory. The purpose of the present paper is to present a self-contained survey of the corresponding developments.  ...  In 2005, Kechris, Pestov and Todorcevic established a surprising correspondence between structural Ramsey theory and topological dynamics.  ...  Below is a list of Fraïssé classes for which it has been applied in order to compute the universal minimal flow of the corresponding automorphism groups.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1412.3254v2">arXiv:1412.3254v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nlaso4d63zdipiltn4hqbii77i">fatcat:nlaso4d63zdipiltn4hqbii77i</a> </span>
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Book Review: Linear orderings

A. M. W. Glass
<span title="1983-07-01">1983</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/iwegpttow5b4vhkhs3scejn5fi" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;
For example, there is a first order sentence of the language of group theory that is satisfied by the automorphism group of a 1-homogeneous linearly ordered set if and only if the linearly ordered set  ...  Moreover, knowledge of permutation groups and linear orderings make this study more tractable-e.g., to prove the undecidability of the theory of lattice-ordered groups, the existence of a 4 generator 1  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0273-0979-1983-15169-8">doi:10.1090/s0273-0979-1983-15169-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/orrosfcx3jbltfezui7vylldn4">fatcat:orrosfcx3jbltfezui7vylldn4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180725034725/http://www.ams.org/journals/bull/1983-09-01/S0273-0979-1983-15169-8/S0273-0979-1983-15169-8.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/01/fa/01fa75f4dcce6e28dcd3041ce7b7efd17a7387b7.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0273-0979-1983-15169-8"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Ramsey property for Boolean algebras with ideals and P(ω_1)/fin [article]

Dana Bartošová
<span title="2013-01-16">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This allows us to compute the universal minimal flow of the group of automorphisms of P(ω_1)/fin, should it be isomorphic to P(ω)/fin or not, and of other quotients of power set algebras.  ...  Taking Fraïssé limits of these classes, we can compute universal minimal flows of groups of homeomorphisms of the Cantor set fixing some closed subsets.  ...  Then NO K (A) is the universal minimal flow for every dense subgroup of Aut(A). Theorem 8 . 8 Let J be a countable linear order and let B be on of the classes B J , B u , B u J .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1212.1529v2">arXiv:1212.1529v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yt3zi4yt6jfhtfw4ngphdeux64">fatcat:yt3zi4yt6jfhtfw4ngphdeux64</a> </span>
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A Survey On Structural Ramsey Theory And Topological Dynamics With The Kechris-Pestov-Todorcevic Correspondence In Mind

Lionel Nguyen Van Thé
<span title="2015-04-15">2015</span> <i title="Zenodo"> Zenodo </i> &nbsp;
As an immediate consequence, it triggered a new interest for structural Ramsey theory. The purpose of the present paper is to present a self-contained survey of the corresponding developments.  ...  A paraître dans Zbornik Radova. (Beograd). Mat. Inst.  ...  Research partially supported by the ANR project Grupo-Loco (ANR-11-JS01-0008)  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5281/zenodo.1471599">doi:10.5281/zenodo.1471599</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hlaz3o7lkzf6dh3sjgxqvfhp7a">fatcat:hlaz3o7lkzf6dh3sjgxqvfhp7a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200305110001/https://zenodo.org/record/1471600/files/article.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/99/07/9907ca26828566a299bfe300698d5d52c446c219.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5281/zenodo.1471599"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> zenodo.org </button> </a>

Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups [article]

A.S. Kechris, V.G. Pestov, S. Todorcevic
<span title="2004-06-24">2004</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study in this paper some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures  ...  Thus the automorphism group of the rationals with the usual ordering and a finite family of subsets each of which is dense and co-dense is extremely amenable.  ...  and ordering properties and uses this to compute the universal minimal flow of the isometry group of the Baire space.  ... 
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