论文标题
带有阿贝里亚谓词的小组方程
Group equations with abelian predicates
论文作者
论文摘要
在本文中，我们开始对具有求解方程的主要类别的Abelian谓词进行系统研究。我们在单词方程式上具有长度约束，更普遍地，将半群的存在理论扩展到群体世界。 我们使用方程式使用解释性来建立模型理论和代数条件，这足以获得不可证明的性能。我们将结果应用于（非亚洲）右角artin组，并表明用阿贝尔谓词求解方程的问题是无法确定的。我们为双曲线基团获得了相同的结果，其双曲线基团至少具有无扭转等级。相比之下，我们证明，在有限的Abelianate化的基团中，可以将问题降低到具有可识别约束的方程式中，因此，这在右角的Coxeter组中是可以决定的，或更普遍地是有限基团的图形产品，以及具有有限余质化的双曲线组。
In this paper we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more generally, on extensions of the existential theory of semigroups, to the world of groups. We use interpretability by equations to establish model-theoretic and algebraic conditions which are sufficient to get undecidability. We apply our results to (non-abelian) right-angled Artin groups, and show that the problem of solving equations with abelian predicates is undecidable for these. We obtain the same result for hyperbolic groups whose abelianisation has torsion-free rank at least two. By contrast, we prove that in groups with finite abelianisation, the problem can be reduced to solving equations with recognisable constraints, and so this is decidable in right-angled Coxeter groups, or more generally, graph products of finite groups, as well as hyperbolic groups with finite abelianisation.
