Related papers: On the (non) superstable part of the free group

A polynomial $f(x)$ over a field $K$ is said to be stable if all its iterates are irreducible over $K$. L. Danielson and B. Fein have shown that over a large class of fields $K$, if $f(x)$ is an irreducible monic binomial, then it is stable…

We show that if $G$ is a sufficiently saturated stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then $G$ is superstable of finite $U$-rank. Combined with recent work of Palacin and Sklinos, we…

A sofic group $G$ is said to be flexibly stable if every sofic approximation to $G$ can converted to a sequence of disjoint unions of Schreier graphs by modifying an asymptotically vanishing proportion of edges. We establish that if…

Let $T$ be a complete, superstable theory with fewer than $2^{\aleph_{0}}$ countable models. Assuming that generic types of infinite, simple groups definable in $T^{eq}$ are sufficiently non-isolated we prove that $\omega^{\omega}$ is the…

We give a description of definable sets $P=(p_1,..., p_m)$ in a free non-abelian group $F$ and in a torsion-free non-elementary hyperbolic group $G$ that follows from our work on the Tarski problems. This answers Malcev's question for $F$.…

This work can be thought as a contribution to the model theory of group extensions. We study the groups G which are interpretable in the disjoint union of two structures (seen as a two-sorted structure). We show that if one of the two…

Let $X$ be a smooth projective surface over an algebraically closed field $k$ of characteristic $p> 0$ with $\Omega_{X}^{1}$ semistable and $\mu(\Omega_{X}^{1})>0$. For any semistable (resp. stable) bundle $W$ of rank $r$, we prove that…

We characterize some classes of finite soluble groups. In particular, we prove that: a finite group $G$ is supersoluble if and only if $G$ has a normal subgroup $D$ such that $G/D$ is supersoluble and $D$ avoids every chief factor of $G$…

In this paper we explore the representation property over sets. This property generalizes constructibility, however is weak enough to enable us to prove that the class of theories $T$ whose models are representable is exactly the class of…

Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the structure induced from N is o-minimal. In particular X is linearly…

We prove that, given $\epsilon>0$ and $k\geq 1$, there is an integer $n$ such that the following holds. Suppose $G$ is a finite group and $A\subseteq G$ is $k$-stable. Then there is a normal subgroup $H\leq G$ of index at most $n$, and a…

We prove that each \omega-categorical, generically stable group is solvable-by-finite.

A new notion of independence relation is given and associated to it, the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence…

Let $A$ be a (not necessarily unital) separable non-elementary simple amenable C*-algebra whose tracial basis may not have finite covering dimension and may not be compact but satisfies certain condition (C). We show that $A$ is ${\cal…

We prove that if $G=(V,E)$ is an $\omega$-stable (respectively, superstable) graph with $\chi(G)>\aleph_0$ (respectively, $2^{\aleph_0}$) then $G$ contains all the finite subgraphs of the shift graph $\text{Sh}_n(\omega)$ for some $n$. We…

A countable discrete group is said to be Frobenius stable if every function from the group to unitary matrices that is "almost multiplicative" in the Frobenius norm is "close" to a unitary representation in the Frobenius norm. The purpose…

We prove that $(\Z,+,0)$ has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank $\omega$. Additionally, our…

We introduce a combinatorial criterion for verifying whether a formula is not the conjunction of an equation and a co-equation. Using this, we give a proof for the nonequationality of the free group. Furthermore, we generalize the latter…

A polynomial $f(x)$ over a field $K$ is called stable if all of its iterates are irreducible over $K$. In this paper we study the stability of trinomials over finite fields. Specially, we show that if $f(x)$ is a trinomial of even degree…

We show that if $G$ is a discrete Abelian group and $A \subset G$ has $\|1_A\|_{B(G)} \leq M$ then $A$ is $O(\exp(\pi M))$-stable in the sense of Terry and Wolf.