论文标题
$ \ Mathbb z^n \ rtimes \ mathbb z $的算术性
Arithmeticity of groups $\mathbb Z^n\rtimes\mathbb Z$
论文作者
论文摘要
我们研究$ \ mathbb z^n \ rtimes_a \ mathbb z $是算术的，其中$ a \ in gl_n（\ mathbb z）$是双曲线和半imimple。首先，在Grunewald-Platonov的作品的基础上，用代数Tori的语言进行了算术的特征。我们用它来证明了几种具体的结果，这些结果将$ \ mathbb z^n \ rtimes_a \ mathbb z $与$ a $的特征多项式的降低属性相关联。我们的工具包括代数托里，有限群体的表示理论，加洛伊斯理论和逆向加洛伊斯问题。
We study when the group $\mathbb Z^n\rtimes_A\mathbb Z$ is arithmetic where $A\in GL_n(\mathbb Z)$ is hyperbolic and semisimple. We begin by giving a characterization of arithmeticity phrased in the language of algebraic tori, building on work of Grunewald-Platonov. We use this to prove several more concrete results that relate the arithmeticity of $\mathbb Z^n\rtimes_A\mathbb Z$ to the reducibility properties of the characteristic polynomial of $A$. Our tools include algebraic tori, representation theory of finite groups, Galois theory, and the inverse Galois problem.
