Diophantine Geometry over Groups X: The Elementary Theory of Free Products of Groups

Diophantine geometry over groups and the elementary theory of free and hyperbolic groups

We study sets of solutions to equations over a free group, projections of such sets, and the structure of elementary sets defined over a free group. The structre theory we obtain enable us to answer some questions of A. Tarski's, and classify those finitely generated groups that are elementary equivalent to a free group. Connections with low dimensional topology, and a generalization to (Gromov) hyperbolic groups will also be discussed.

Diophantine Geometry over Groups VIII: Stability

This paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of definable sets over free and hyperbolic groups. In the eighth paper we use a modification of the sieve procedure, that was presented as part of the quantifier elimination process, to prove that free and torsion-free (Gromov) hyperbolic groups are stable.

丢番图几何与群论 IX：包络与虚元素

Diophantine Geometry over Groups IX: Envelopes and Imaginaries

本文通过将可定义集关联到丢番图包络，并利用对偶极限群研究可定义等价关系，从而对自由群与双曲群中的虚元素进行分类。

在各向同性可约群方案中的丢番图问题

The Diophantine problem in isotropic reductive groups

本文研究了各向同性可约群方案中的丢番图问题，证明了基环与群点的丢番图问题等价，并计算了相关子群的中心化器和正规化子。

On Profinite Hyperbolicity and Diophantine Geometry

In this note, we explore the notion of hyperbolicity of topologically finitely generated profinite groups. Some applications to diophantine geometry are suggested and we try to reformulate certain problems in diophantine geometry in terms of hyperbolic profinite groups. Then, we introduce many occasions in which Galois groups are free profinite and try to explore implications of this condition in the world of diophantine geometry. In particular, we prove that, Grothendieck's "section conjecture" plus Shafarevich's "freeness conjecture" imply that hyperbolic curves have infinitely many solution…

Chevalley群中的丢番图问题

The Diophantine problem in Chevalley groups

本文证明任意含1交换环上Chevalley群中的丢番图问题与环本身的丢番图问题Karp等价，并由此得到若干模型论推论。