Bohr compactifications of groups and rings

Schur超滤子与拓扑群的Bohr紧化

Schur ultrafilters and Bohr compactifications of topological groups

本文利用Schur超滤子与Stone-Čech紧化的代数结构，给出了拓扑群Bohr紧化的新描述，并证明了chart群为拓扑群等价于其每个Schur超滤子收敛到单位元。

Remarks on compactifications of pseudofinite groups

We discuss the Bohr compactification of a pseudofinite group, motivated by a question of Boris Zilber. Basically referring to results in the literature we point out (i) the Bohr compactification of an ultraproduct of finite simple groups is trivial, and (ii) the "definable" Bohr compactification of any pseudofinite group G, relative to a nonstandard model of set theory in which it is definable, is commutative-by-profinite.

Generalized Bohr compactification and model-theoretic connected components

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 …

Definable topological dynamics

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 $Δ$-definable $G$-flows for any fixed set $Δ$ of formulas of an…

On compactifications and the topological dynamics of definable groups

We discuss definable compactifications and topological dynamics. For G a group definable in some structure M, we define notions of "definable" compactification of G and "definable" action of G on a compact space X (definable G-flow), where the latter is under a definability of types assumption on M. We describe the universal definable compactification of G as G*/G*00_M and the universal definable G-ambit as the type space S_{G}(M). We also prove existence and uniqueness of "universal minimal definable G-flows", and discuss issues of amenability and extreme amenability in this definable categor…

Structure and regularity for subsets of groups with finite VC-dimension

Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen $ε>0$, describe the structure of $A$ and behave regularly with respect to translates of $A$. For the subclass of groups with uniformly fixed finite exponent $r$, these algebraic objects are normal subgroups with index bounded in terms of $k$, $r$, and $ε$. For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model theoret…