Minimal bounded index subgroup for dependent theories

Universally measurable subgroups of countable index

It is proved that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group $S_\infty$ is continuous. It is also shown that a universally measurable homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous.

An axiomatizable profinite group with infinitely many open subgroups of index 2

We show that a profinite group with the same first-order theory as the direct product over all odd primes $p$ of the dihedral group of order $2p$, is necessarily isomorphic to this direct product.

A Fitting Theorem for Simple Theories

The Fitting subgroup of a type-definable group in a simple theory is relatively definable and nilpotent. Moreover, the Fitting subgroup of a supersimple hyperdefinable group has a normal hyperdefinable nilpotent subgroup of bounded index, and is itself of bounded index in a hyperdefinable subgroup.

On expansions of non-abelian free groups by cosets of a finite index subgroup

Let $F$ be a finitely generated non-abelian free group and $Q$ a finite quotient. Denote by $L_Q$ the language obtained by adding unary predicates $P_q$, $q\in Q$ to the language of groups. Using a slight generalization of some of the techniques involved in Zlil Sela's solution to Tarskiś problem on the elementary theory of non-abelian free groups, we provide a few basic results on the validity of first order entences in the $L_Q$-expansion of $F$ in which every $P_q$ is interpreted as the preimage of $q$ in $F$. In particular we prove an analogous result to Sela's generalization of Merzlyakov…

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。