论文标题
关于某些单层群中的双磷酸问题
On the Diophantine problem in some one-relator groups
论文作者
论文摘要
我们研究了养分剂问题，即为单宗教群体的某些家庭解决方程式系统的决策问题，并提供了一些背景，以了解为什么这个问题引起了人们的关注。所使用的方法主要是Reidemeister-Schreier方法，以及Dahmani＆Guirardel和Ciobanu，Holt＆Rees的一般结果，涉及一般类别中的二芬太汀问题的可决定性。首先，我们通过证明具有定义关系$ a^mb^n = 1 $的单级式群体的示例实际上是所有$ m，n \ geq 0 $的直接产物，因此在此类组中得出了二苯胺问题的确定性。作为推论，我们可以获得在任何圆环结组中都可以决定养生问题的问题。其次，我们研究了两个及单层群组$ g_ {m，n} $，并定义关系a换向器$ [a^m，b^n] = 1 $，其中$ m，n \ geq 1 $。在此过程中，我们定义并研究了与右角artin组（RAAGS）有关的自然类（Rabsags）。我们将$ g_ {m，n} $组中的双磷酸问题减少到几乎是某些rabsags的组中的diophantine问题。作为我们方法的推论，我们表明，在组$ g_ {2,2} $的组中，submonoid成员资格问题与单个定义关系$ [a^2，b^2] = 1 $。我们使用灰色和霍伊（Grey＆Howie）的最新分类，对单层群的RAAG亚组进行了分类，以对某些Rabsags的RAAG亚组进行分类，显示了单余理理论对该领域的潜在有用性。最后，我们定义和研究Newman $ \ operatotorname {ng}（p，q）$，是$（p+1）$ - 生成的单式群体群体，可以推广可解决的baumslag- solaritar群体。我们表明所有这些组都是双曲线，从而得出结论他们的二磷剂问题的可决定性。
We study the Diophantine problem, i.e. the decision problem of solving systems of equations, for some families of one-relator groups, and provide some background for why this problem is of interest. The method used is primarily the Reidemeister--Schreier method, together with general recent results by Dahmani & Guirardel and Ciobanu, Holt & Rees on the decidability of the Diophantine problem in general classes of groups. First, we give a sample of the methods of the article by proving that the one-relator group with defining relation $a^mb^n = 1$ is virtually a direct product of hyperbolic groups for all $m, n \geq 0$, and thus conclude decidability of the Diophantine problem in such groups. As a corollary, we obtain that the Diophantine problem is decidable in any torus knot group. Second, we study the two-generator, one-relator groups $G_{m,n}$ with defining relation a commutator $[a^m, b^n] = 1$, where $m, n \geq 1$. In doing so, we define and study a natural class of groups (RABSAGs), related to right-angled Artin groups (RAAGs). We reduce the Diophantine problem in the groups $G_{m,n}$ to the Diophantine problem in groups which are virtually certain RABSAGs. As a corollary of our methods, we show that the submonoid membership problem is undecidable in the group $G_{2,2}$ with the single defining relation $[a^2, b^2] = 1$. We use the recent classification by Gray & Howie of RAAG subgroups of one-relator groups to classify the RAAG subgroups of some RABSAGs, showing the potential usefulness of one-relator theory to this area. Finally, we define and study Newman groups $\operatorname{NG}(p,q)$, which are $(p+1)$-generated one-relator groups generalising the solvable Baumslag--Solitar groups. We show that all such groups are hyperbolic, and thereby also conclude decidability of their Diophantine problem.
