Group Equations With Abelian Predicates
Group Equations With Abelian Predicates
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带有阿贝尔谓词的群方程
DOI：
10.1093/imrn/rnad179
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发表时间：
2023
影响因子：
1
通讯作者：
Ciobanu L
中科院分区：
文献类型：
--
作者：
Ciobanu L
在这篇文章中，我们开始在主要的群类中系统地研究具有交换谓词的群方程，其中方程是可以求解的。我们把带长度约束的字方程的工作扩展到群的世界，更广泛地说，扩展了半群的存在理论。我们使用方程的可解释性来建立模型论和代数条件，这些条件足以得到不可判定。我们将我们的结果应用于(非交换的)直角Artin群，并证明了对于这些群，求解带有交换谓词的方程的问题是不可判定的。对于双曲群，我们得到了相同的结果，这些双曲群的无挠阶数至少为2。相反，我们证明了在有限简化的群中，问题可以归结为求解具有可识别约束的方程，因此这在直角Coxeter群中是可判定的，或者更一般地，有限简化的群的图积，以及有限简化的双曲群。
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.