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Register a new userAmenability and ergodic properties of topological groups: from Bogolyubov onwards
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The purpose of this expository article is to revisit the notions of amenability and ergodicity, and to point out that they appear for topological groups that are not necessarily locally compact in articles by Bogolyubov (1939), Fomin (1950), Dixmier (1950), and Rickert (1967).
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Generalizing Block and Weinbergers characterization of amenability we introduce the notion of uniformly finite homology for a group action on a compact space and use it to give a homological characterization of topological amenability for actions. By considering the case of the natural action of $G$ on its Stone-vCech compactification we obtain a homological characterization of exactness of the group, answering a question of Nigel Higson.
We prove that the alternating group of a topologically free action of a countably infinite group $Gamma$ on the Cantor set has the property that all of its $ell^2$-Betti numbers vanish and, in the case that $Gamma$ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable $Gamma$ that there are many such alternating groups which are simple, finitely generated, and C$^*$-simple. The device for establishing nonisomorphism among these examples is a topological version of Austins result on the invariance of measure entropy under bounded orbit equivalence.
In this paper, we investigate algebraic and topological properties of the Riordan groups over finite fields. These groups provide a new class of topologically finitely generated profinite groups with finite width. We also introduce, characterize index-subgroups of our Riordan groups, and finally we show exactly the range of Hausdorff dimensions of these groups. The latter results are analogous to the work of Barnea and Klopsch for the Nottingham groups.
We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness, ...). For example, if $M$ is a countable, $omega$-categorical structure and $Aut(M)$ is amenable, as a topological group, then the Lascar Galois group $Gal_{L}(T)$ of the theory $T$ of $M$ is compact, Hausdorff (also over any finite set of parameters), that is $T$ is G-compact. An essentially special case is that if $Aut(M)$ is extremely amenable, then $Gal_{L}(T)$ is trivial, so, by a theorem of Lascar, the theory $T$ can be recovered from its category $Mod(T)$ of models. On the side of definable groups, we prove for example that if $G$ is definable in a model $M$, and $G$ is definably amenable, then the connected components ${G^{*}}^{00}_{M}$ and ${G^{*}}^{000}_{M}$ coincide, answering positively a question from an earlier paper of the authors. We also take the opportunity to further develop the model-theoretic approach to topological dynamics, obtaining for example some new invariants for topological groups, as well as allowing a uniform approach to the theorems above and the various categories.
We construct locally compact groups with no non-trivial Invariant Random Subgroups and no non-trivial Uniformly Recurrent Subgroups.
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