Arithmetic methods for finitely generated matrix groups

有限生成矩阵群的算术方法

• 批准号: 238042377
• 负责人: Professorin Dr. Gabriele Nebe
• 金额: --
• 依托单位: Lehrstuhl für Algebra und Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/238042377?language=en
• 关键词: Arithmetic; methods; finitely; generated; matrix

The membership problem is in general undecidable for finitely generated matrix groups over infinite fields. Nevertheless, certain, naturally arising matrix groups can be handled algorithmically. Examples are given by normalizers of finite matrix group, automorphism groups of hyperbolic Lattices and groups generated by a certain family of maximal finite matrix groups. The latter naturally act on Bruhat-Tits buildings of p-adic classical groups which can be used to obtain a structural description of the group as well as algorithmic methods, e.g. a membership test. To analyze subgroups of PSL2 one may apply the natural action on hyperbolic spaces. We want to develop general purpose algorithms for a geometric reduction a la Aschbacher also for infinite fields. Arithmetic methods (invariant lattices, enveloping orders) will lead to the construction of invariants of finitely generated matrix groups, like finite factor groups and p-adic completions.

对于无限域上有限生成的矩阵群，隶属度问题通常是不可判定的。然而，某些自然产生的矩阵组可以通过算法进行处理。例子是有限矩阵群的归一化器、双曲格的自同构群以及由最大有限矩阵群的某个族生成的群。后者自然地作用于 p-adic 经典群的 Bruhat-Tits 建筑物，可用于获得群的结构描述以及算法方法，例如会员资格测试。为了分析 PSL2 的子群，可以应用双曲空间上的自然作用。我们希望开发通用算法，用于几何简化，就像阿施巴赫（Aschbacher）一样，也适用于无限域。算术方法（不变格、包络阶）将导致有限生成矩阵群的不变量的构造，例如有限因子群和 p-adic 完成。