Arithmetic methods for finitely generated matrix groups

对于无限域上的有限生成矩阵群，隶属度问题一般是不可判定的。然而，某些自然产生的矩阵群可以通过算法来处理。给出了有限矩阵群的正规化子、双曲格的自同构群以及由某一族极大有限矩阵群生成的群的例子。后者自然而然地作用于p-进经典群的Bruhat-Tits构造物，可用于获得群的结构描述以及算法方法，例如成员关系检验。为了分析PSL2的子群，人们可以在双曲空间上应用自然作用。我们想要开发几何归约的通用算法，就像Aschbacher一样，也适用于无穷域。算术方法(不变格、包络序)将导致构造有限生成矩阵群的不变量，如有限因子群和p-进补。

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.