41 research outputs found

    Automorphism groups and Ramsey properties of sparse graphs

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    We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todor\v{c}evi\'c correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory. Resolving one of the open questions in the area, we show that Hrushovski's example of an ω\omega-categorical sparse graph has no ω\omega-categorical expansion with extremely amenable automorphism group

    Amenability, connected components, and definable actions

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    We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if GG is an amenable topological group, then the Bohr compactification of GG coincides with a certain ``weak Bohr compactification'' introduced in [24]. In other words, the conclusion says that certain connected components of GG coincide: Gtopo00=Gtopo000G^{00}_{topo} = G^{000}_{topo}. We also prove wide generalizations of this result, implying in particular its extension to a ``definable-topological'' context, confirming the main conjectures from [24]. We also introduce \bigvee-definable group topologies on a given \emptyset-definable group GG (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of GG implies (under some assumption) that cl(GM00)=cl(GM000)cl(G^{00}_M) = cl(G^{000}_M) for any model MM. Thirdly, we give an example of a \emptyset-definable approximate subgroup XX in a saturated extension of the group F2×Z\mathbb{F}_2 \times \mathbb{Z} in a suitable language (where F2\mathbb{F}_2 is the free group in 2-generators) for which the \bigvee-definable group H:=XH:=\langle X \rangle contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) ``model'' exists for each approximate subgroup does not work in general (they proved in [29] that it works for definably amenable approximate subgroups).Comment: Version 3 contains the material in Sections 2, 3, and 5 of version 1. Following the advice of editors and referees we have divided version 1 into two papers, version 3 being the first. The second paper is entitled "On first order amenability

    Canonical functions: a proof via topological dynamics

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    Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties. In this short note, we present a proof of the existence of canonical functions in certain sets using topological dynamics, providing a shorter alternative to the original combinatorial argument. We moreover present equivalent algebraic characterisations of canonicity

    Ramsey expansions of metrically homogeneous graphs

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    We discuss the Ramsey property, the existence of a stationary independence relation and the coherent extension property for partial isometries (coherent EPPA) for all classes of metrically homogeneous graphs from Cherlin's catalogue, which is conjectured to include all such structures. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and the coherent EPPA. Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's classification programme of Ramsey classes and as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion (or lift or ordering) property, EPPA and the existence of a stationary independence relation. At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin's classes. The existence of such a "completion algorithm" then allows us to apply several strong results in the areas that imply EPPA and respectively the Ramsey property. The main results have numerous corollaries on the automorphism groups of the Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor revisio

    Aspects of the topological dynamics of sparse graph automorphism groups

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    We examine sparse graph automorphism groups from the perspective of the Kechris-Pestov-Todorčević (KPT) correspondence. The sparse graphs that we discuss are Hrushovski constructions: we consider the 'ab initio’ Hrushovski construction M_0, the Fraïssé limit of the class of 2-sparse graphs with self-sufficient closure; M_1, a simplified version of M_0; and the ω-categorical Hrushovski construction M_F. We prove a series of results that show that the automorphism groups of these Hrushovski constructions demonstrate very different behaviour to previous classes studied in the KPT context. Extending results of Evans, Hubička and Nešetřil, we show that Aut(M_0) has no coprecompact amenable subgroup. We investigate the fixed points on type spaces property, a weakening of extreme amenability, and show that for a particular choice of control function F, Aut(M_F) does not have any closed oligomorphic subgroup with this property. Next we consider the Aut(M_1)-flow of linear orders on M_1, and show that minimal subflows of this have all Aut(M_1)-orbits meagre. We give partial analogous results for the Aut(M_0)-flow of linear orders on M_0, and find the universal minimal flow of the automorphism group of the “dimension 0” part of M_0.Open Acces

    Structures métriques et leurs groupes d’automorphismes : reconstruction, homogénéité, moyennabilité et continuité automatique

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    This thesis focuses on the study of Polish groups seen as automorphism groups of metric structures. The observation that every non-archimedean Polish group is the automorphism group of an ultrahomogeneous countable structure has indeed led to fruitful interactions between group theory and model theory. In the framework of metric model theory, introduced by Ben Yaacov, Henson and Usvyastov, this correspondence has been extended to all Polish groups by Melleray. In this thesis, we study various facets of this correspondence. The relationship between a structure and its automorphism group is particularly close in the setting of ℵ0-categorical structures. Indeed, the Ahlbrandt-Ziegler reconstruction theorem allows one to recover an ℵ0-categorical structure, up to bi-interpretability, from its automorphism group. In a joint work with Itai Ben Yaacov, we generalize this result to separably categorical metric structures. Besides, ultrahomogeneous countable structures have the advantage of being completely determined by their finitely generated substructures. In particular, this enabled Moore to give a combinatorial characterization of amenability for nonarchimedean Polish groups. We extend this characterization to all Polish groups and we deduce that amenability is a Gδ condition. Still in a reconstruction perspective, we are interested in the automatic continuity property for Polish groups. Sabok and Malicki introduced conditions of a combinatorial nature on an ultrahomogeneous metric structure that imply the automatic continuity property for its automorphism group. We show that these conditions carry to countable powers, which leads to the groups Aut(μ)N, U(l2)N and Iso(U)N satisfying the automatic continuity property. Those conditions are a weakening of the property of having ample generics. In a joint work with Francois Le Maitre, we exhibit the first examples of connected groups with ample generics, which answers a question of Kechris and Rosendal. Finally, in a joint work with Isabel Muller and Aristotelis Panagiotopoulos, we study the relative homogeneity of substructures in an ultrahomogeneous countable structure. We characterize it completely by a property of the types over the substructures: being determined by a finite setCette thèse porte sur l'étude des groupes polonais vus comme groupes d'automorphismes de structures métriques. L'observation que tout groupe polonais non archimédien est le groupe d'automorphismes d'une structure dénombrable ultra homogène a en effet mené à des interactions fructueuses entre la théorie des groupes et la théorie des modèles. Dans le cadre de la théorie des modèles métriques, introduite par Ben Yaacov, Henson et Usvyatsov, cette correspondance a été étendue par Melleray à tous les groupes polonais. Dans cette thèse, nous étudions diverses facettes de cette correspondance. Le lien entre une structure et son groupe d automorphismes est particulièrement étroit dans le cadre des structures ℵ0-categoriques. En effet, le théorème de reconstruction d'Ahlbrandt-Ziegler permet de retrouver une structure ℵ0-categorique, à bi-interprètabilité près, à partir de son groupe d'automorphismes. Dans un travail en commun avec Itai Ben Yaacov, nous généralisons ce résultat aux structures métriques separablement catégoriques. Les structures dénombrables ultra homogènes ont de plus l avantage d'être complètement déterminées par leurs sous-structures finiment engendrées. Cela a notamment permis a Moore de donner une caractérisation combinatoire de la moyennabilité des groupes polonais non archimédiens. Nous étendons cette caractérisation à tous les groupes polonais et nous en déduisons que la moyennabilite est une condition Gδ. Toujours dans une optique de reconstruction, nous nous intéressons à la propriété de continuité automatique pour les groupes polonais. Sabok et Malicki ont introduit des conditions de nature combinatoire sur une structure métrique ultra homogène qui impliquent la propriété de continuité automatique pour son groupe d'automorphismes. Nous montrons que ces conditions passent à la puissance dénombrable, ce qui a pour conséquence que les groupes Aut(μ)N, U(l2)N et Iso(U)N satisfont la propriété de continuité automatique. Ces conditions sont un affaiblissement du fait d'avoir des amples génériques. Dans un travail en commun avec Francois Le Maitre, nous exhibons les premiers exemples de groupes connexes qui ont des amples génériques, ce qui répond à une question de Kechris et Rosenda

    Topological properties of Wazewski dendrite groups

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    Homeomorphism groups of generalized Wa\.zewski dendrites act on the infinite countable set of branch points of the dendrite and thus have a nice Polish topology. In this paper, we study them in the light of this Polish topology. The group of the universal Wa\.zewski dendrite DD_\infty is more characteristic than the others because it is the unique one with a dense conjugacy class. For this group GG_\infty, we show some of its topological properties like existence of a comeager conjugacy class, the Steinhaus property, automatic continuity and the small index subgroup property. Moreover, we identify the universal minimal flow of GG_\infty. This allows us to prove that point-stabilizers in GG_\infty are amenable and to describe the universal Furstenberg boundary of GG_\infty.Comment: Slight modifications about the expositio
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