Arithmetic methods for finitely generated matrix groups

• Applicant: Professorin Dr. Gabriele Nebe
• Subject Area: Mathematics
• Term: from 2013 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 238042377

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1489: Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory