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Register a new userAmenability, 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 $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain ``weak Bohr compactification introduced in [24]. In other words, the conclusion says that certain connected components of $G$ coincide: $G^{00}_{topo} = G^{000}_{topo}$. 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 $G$ (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 $G$ implies (under some assumption) that $cl(G^{00}_M) = cl(G^{000}_M)$ for any model $M$. Thirdly, we give an example of a $emptyset$-definable approximate subgroup $X$ in a saturated extension of the group $mathbb{F}_2 times mathbb{Z}$ in a suitable language (where $mathbb{F}_2$ is the free group in 2-generators) for which the $bigvee$-definable group $H:=langle X rangle$ 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).
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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.
For a group $G$ definable in a first order structure $M$ we develop basic topological dynamics in the category of definable $G$-flows. In particular, we give a description of the universal definable $G$-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of $G$, that is to the quotient $G^*/{G^*}^{00}_M$ (where $G^*$ is the interpretation of $G$ in a monster model). More generally, we obtain these results locally, i.e. in the category of $Delta$-definable $G$-flows for any fixed set $Delta$ of formulas of an appropriate form. In particular, we define local connected components ${G^*}^{00}_{Delta,M}$ and ${G^*}^{000}_{Delta,M}$, and show that $G^*/{G^*}^{00}_{Delta,M}$ is the $Delta$-definable Bohr compactification of $G$. We also note that some deeper arguments from the topological dynamics in the category of externally definable $G$-flows can be adapted to the definable context, showing for example that our epimorphism from the Ellis group to the $Delta$-definable Bohr compactification factors naturally yielding a continuous epimorphism from the $Delta$-definable generalized Bohr compactification to the $Delta$-definable Bohr compactification of $G$. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.
We show that the universal minimimal proximal flow and the universal minimal strongly proximal flow of a discrete group can be realized as the Stone spaces of translation invariant Boolean algebras of subsets of the group satisfying a higher order notion of syndeticity. We establish algebraic, combinatorial and topological dynamical characterizations of these subsets that we use to obtain new necessary and sufficient conditions for strong amenability and amenability. We also characterize dense orbit sets, answering a question of Glasner, Tsankov, Weiss and Zucker.
For a group $G$ first order definable in a structure $M$, we continue the study of the definable topological dynamics of $G$. The special case when all subsets of $G$ are definable in the given structure $M$ is simply the usual topological dynamics of the discrete group $G$; in particular, in this case, the words externally definable and definable can be removed in the results described below. Here we consider the mutual interactions of three notions or objects: a certain model-theoretic invariant $G^{*}/(G^{*})^{000}_{M}$ of $G$, which appears to be new in the classical discrete case and of which we give a direct description in the paper; the [externally definable] generalized Bohr compactification of $G$; [externally definable] strong amenability. Among other things, we essentially prove: (i) The new invariant $G^{*}/(G^{*})^{000}_{M}$ lies in between the externally definable generalized Bohr compactification and the definable Bohr compactification, and these all coincide when $G$ is definably strongly amenable and all types in $S_G(M)$ are definable, (ii) the kernel of the surjective homomorphism from $G^*/(G^*)^{000}_M$ to the definable Bohr compactification has naturally the structure of the quotient of a compact (Hausdorff) group by a dense normal subgroup, and (iii) when $Th(M)$ is NIP, then $G$ is [externally] definably amenable iff it is externally definably strongly amenable. In the situation when all types in $S_G(M)$ are definable, one can just work with the definable (instead of externally definable) objects in the above results.
We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmueller geodesic flow restricted to the hyperelliptic connected components, settling a question of Farb. We stress that all proofs and computations are performed without the help of a computer. As a byproduct, our methods give a way to describe the bottom of the lengths spectrum of the hyperelliptic components.
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