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Automorphism groups of right-angled buildings: simplicity and local splittings

Pierre-Emmanuel Caprace (2014)

Fundamenta Mathematicae

We show that the group of type-preserving automorphisms of any irreducible semiregular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated abstractly simple locally compact groups. Specialising to appropriate cases, we obtain examples of such simple groups that are locally indecomposable, but have locally normal subgroups decomposing non-trivially as direct products, all of whose factors are locally normal.

Combinatorial and group-theoretic compactifications of buildings

Pierre-Emmanuel Caprace, Jean Lécureux (2011)

Annales de l’institut Fourier

Let X be a building of arbitrary type. A compactification 𝒞 sph ( X ) of the set Res sph ( X ) of spherical residues of X is introduced. We prove that it coincides with the horofunction compactification of Res sph ( X ) endowed with a natural combinatorial distance which we call the root-distance. Points of 𝒞 sph ( X ) admit amenable stabilisers in Aut ( X ) and conversely, any amenable subgroup virtually fixes a point in 𝒞 sph ( X ) . In addition, it is shown that, provided Aut ( X ) is transitive enough, this compactification also coincides with the group-theoretic...

Fraïssé structures and a conjecture of Furstenberg

Dana Bartošová, Andy Zucker (2019)

Commentationes Mathematicae Universitatis Carolinae

We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between S ( G ) , the Samuel compactification, and E ( M ( G ) ) , the enveloping semigroup of the universal minimal flow. We resolve Furstenberg’s problem for several automorphism groups and give a detailed study in the case of G = S , leading us to define and investigate several new types...

Lie group structures on groups of diffeomorphisms and applications to CR manifolds

M. Salah Baouendi, Linda Preiss Rothschild, Jörg Winkelmann, Dimitri Zaitsev (2004)

Annales de l’institut Fourier

We give general sufficient conditions to guarantee that a given subgroup of the group of diffeomorphisms of a smooth or real-analytic manifold has a compatible Lie group structure. These results, together with recent work concerning jet parametrization and complete systems for CR automorphisms, are then applied to determine when the global CR automorphism group of a CR manifold is a Lie group in an appropriate topology.

Minimal actions of homeomorphism groups

Yonatan Gutman (2008)

Fundamenta Mathematicae

Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on Φ 2 2 X , the space of maximal chains in 2 X , equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on U H o m e o ( X ) , the universal minimal space of Homeo(X), is not transitive (improving...

More on the Kechris-Pestov-Todorcevic correspondence: Precompact expansions

L. Nguyen Van Thé (2013)

Fundamenta Mathematicae

In 2005, the paper [KPT05] by Kechris, Pestov and Todorcevic provided a powerful tool to compute an invariant of topological groups known as the universal minimal flow. This immediately led to an explicit representation of this invariant in many concrete cases. However, in some particular situations, the framework of [KPT05] does not allow one to perform the computation directly, but only after a slight modification of the original argument. The purpose of the present paper is to supplement [KPT05]...

On the dynamics of (left) orderable groups

Andrés Navas (2010)

Annales de l’institut Fourier

We develop dynamical methods for studying left-orderable groups as well as the spaces of orderings associated to them. We give new and elementary proofs of theorems by Linnell (if a left-orderable group has infinitely many orderings, then it has uncountably many) and McCleary (the space of orderings of the free group is a Cantor set). We show that this last result also holds for countable torsion-free nilpotent groups which are not rank-one Abelian. Finally, we apply our methods to the case of braid...

Random orderings and unique ergodicity of automorphism groups

Omer Angel, Alexander S. Kechris, Russell Lyons (2014)

Journal of the European Mathematical Society

We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner andWeiss’s example of the group of all permutations...

Stabilizers of closed sets in the Urysohn space

Julien Melleray (2006)

Fundamenta Mathematicae

Building on earlier work of Katětov, Uspenskij proved in [8] that the group of isometries of Urysohn's universal metric space 𝕌, endowed with the pointwise convergence topology, is a universal Polish group (i.e. it contains an isomorphic copy of any Polish group). Answering a question of Gao and Kechris, we prove here the following, more precise result: for any Polish group G, there exists a closed subset F of 𝕌 such that G is topologically isomorphic to the group of isometries of 𝕌 which map...

Topological dynamics of unordered Ramsey structures

Moritz Müller, András Pongrácz (2015)

Fundamenta Mathematicae

We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow M ( G ) of the automorphism group G of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of M ( G ) . We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and G is amenable, then M ( G ) admits a unique invariant...

Transition operators on co-compact G-spaces.

Laurent Saloff-Coste, Wolfgang Woess (2006)

Revista Matemática Iberoamericana

We develop methods for studying transition operators on metric spaces that are invariant under a co-compact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce reduced transition operators on the compact factor space whose norms and spectral radii are upper bounds for the Lp-norms and spectral radii of the original operator. If the group is amenable then the spectral radii of the original and reduced operators coincide,...

Un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert

Mickaël Crampon, Ludovic Marquis (2013)

Annales mathématiques Blaise Pascal

On montre un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert. Plus précisément, en toute dimension n , il existe une constante ε n > 0 telle que, pour tout ouvert proprement convexe Ω , pour tout point x Ω , tout groupe discret engendré par un nombre fini d’automorphismes de Ω qui déplacent le point x de moins de ε n est virtuellement nilpotent.

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