论文标题
Coxeter组的一阶方面
First-Order Aspects of Coxeter Groups
论文作者
论文摘要
我们奠定了Coxeter组的一阶模型理论的基础。首先，除了$ 2 $ - 球面的非婚外案例（我们打开）之外，我们表征了有限等级的超级巨星Coxeter群体，我们证明这本质上是仿射类型的Coxeter组。其次，我们表征有限等级的Coxeter组，该组是域，这是代数几何学理论中的一个核心假设，而在许多方面（例如，$λ$稳定性）将给定的Coxeter系统的模型理论降低到其可相关的不可删除成分的模型理论。 在本文的第二部分中，我们将右角的Coxeter组（RACGS）和2美元的Coxetrical Coxeter组中的特定确定性问题转移到了特定的确定性问题。在这方面，首先，我们证明有限等级的岩石没有适当的基本亚组为Coxeter组，并进一步证明，反射独立的基本组根本没有适当的基本亚组。其次，我们证明，如果有限地生成$ w $的$ s $ sum-subliminity的单型$ sim（w，s）$，那么$ w $是其理论的主要模型。第三，我们证明在有限等级的反射独立赛车中，Coxeter元素是类型确定的。然后，我们转移到$ 2 $ - 球面的Coxeter组，证明如果$（w，s）$是不可约定的，甚至$ 2 $ spherical，甚至不呈现，那么$ w $是其理论的主要模型，并且如果$w_γ$和$w_θ$在上一句话中一样，那么$w_γ$是$w_γ$ nim $w_θ$，而仅是$w_θ$，那么如果大多数$ 2 $ - 球面的Coxeter组中的基本等效问题。 在本文的最后一部分中，我们重点介绍了Coxeter群体理论的反射长度概念的模型理论应用，特别是证明了仿射Coxeter基团没有连接。
We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the $2$-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show to be essentially the Coxeter groups of affine type. Secondly, we characterize the Coxeter groups of finite rank which are domains, a central assumption in the theory of algebraic geometry over groups, which in many respects (e.g. $λ$-stability) reduces the model theory of a given Coxeter system to the model theory of its associated irreducible components. In the second part of the paper we move to specific definability questions in right-angled Coxeter groups (RACGs) and $2$-spherical Coxeter groups. In this respect, firstly, we prove that RACGs of finite rank do not have proper elementary subgroups which are Coxeter groups, and prove further that reflection independent ones do not have proper elementary subgroups at all. Secondly, we prove that if the monoid $Sim(W, S)$ of $S$-self-similarities of $W$ is finitely generated, then $W$ is a prime model of its theory. Thirdly, we prove that in reflection independent RACGs of finite rank the Coxeter elements are type-determined. We then move to $2$-spherical Coxeter groups, proving that if $(W, S)$ is irreducible, $2$-spherical even and not affine, then $W$ is a prime model of its theory, and that if $W_Γ$ and $W_Θ$ are as in the previous sentence, then $W_Γ$ is elementary equivalent to $W_Θ$ if and only if $Γ\cong Θ$, thus solving the elementary equivalence problem for most of the $2$-spherical Coxeter groups. In the last part of the paper we focus on model theoretic applications of the notion of reflection length from Coxeter group theory, proving in particular that affine Coxeter groups are not connected.
