Stable group theory and approximate subgroups

On maximal stable quotients of definable groups in NIP theories

For G a group definable in an NIP theory we prove that there is a smallest type-definable subgroup H of G such that the quotient G/H is stable. This generalizes the existence of G^00, the smallest type-definable subgroup of G of bounded index.

关于度量近似子群

On metric approximate subgroups

本文将离散近似子群的 Lie 模型定理推广到双不变度量群，在有限覆盖条件下证明度量近似子群可由典范 Lie 群解释，并推出有界指数群中的近子群结论。

具有有界VC维的近似子群

Approximate subgroups with bounded VC-dimension

本文将Breuillard-Green-Tao近似群结构理论与驯服算术正则性方法结合，证明了任意群中具有小三倍增长和有界VC维的有限子集可分解为有界多个有界秩步长的陪集零进步的平移并集，并推广了有限稳定集的相关结果。

The automorphism group of a saturated model has a large dense free subgroup

We prove that for a stable theory $T,$ if $M$ is a saturated model of $T$ of cardinality $λ$ where $λ> \big|T\big|,$ then $Aut(M)$ has a dense free subgroup on $2^λ$ generators. This affirms a conjecture of Hodges.

Solvability in groups with a chain condition on uniformly definable subgroups

We prove a structure theorem for periodic locally soluble groups satisfying a chain condition on intersections of relatively uniformly definable subgroups using results from the theory of stable groups. The result in particular shows that these groups are soluble, thus giving a model-theoretic and much simpler proof of a special case of a theorem of Bryant and Hartley on the solubility of periodic locally soluble groups satisfying the minimal condition on centralizers.

On stable quotients

We solve two problems from the paper "On maximal stable quotients of definable groups in NIP theories" by M. Haskel and A. Pillay, which concern maximal stable quotients of groups type-definable in NIP theories. The first result says that if $G$ is a type-definable group in a distal theory, then $G^{st}=G^{00}$ (where $G^{st}$ is the smallest type-definable subgroup with $G/G^{st}$ stable, and $G^{00}$ is the smallest type-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from $T$ to the hyperimaginary expansion $T^{heq}$. The second r…

有限群的模型论 tame 性

Model-theoretic Tameness in finite extensions of groups

研究显示，ω-稳定群的有限指数扩展和子群可能在模型论上野性，证明存在这样的群G，使得任意可数一阶结构均可在G的有限指数扩展和子群中解释。

A Note on Generic Types

In a stable abelian group, we characterize generic types of cosets of type-definable subgroups.

Locally definable subgroups of semialgebraic groups

We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $Σ_{i=1}^m(X-X)$ is an approximate subgroup of $G$.

Algebraic description of limit models in classes of abelian groups

We study limit models in the class of abelian groups with the subgroup relation and in the class of torsion-free abelian groups with the pure subgroup relation. We show: $\textbf{Theorem}$ (1) If $G$ is a limit model of cardinality $λ$ in the class of abelian groups with the subgroup relation, then $G \cong (\oplus_λ\mathbb{Q}) \oplus \oplus_{p \text{ prime}} (\oplus_λ \mathbb{Z}(p^\infty))$. (2) If $G$ is a limit model of cardinality $λ$ in the class of torsion-free abelian groups with the pure subgroup relation, then: * If the length of the chain has uncountable cofinality, then $ G \cong (\…