Root graded groups

Abstract

A root graded group is a group containing a family of subgroups that is indexed by a root system and satisfies certain commutation relations. The standard examples are Chevalley groups over rings. The definition of a root grading of a group is inspired by the corresponding notion for Lie algebras for which there are classification results due to Berman, Moody, Benkart and Zelmanov from the 1990s. Much less is known in the group case.

In my talk I will address the classification problem for root graded groups and its connection to the theory of buildings. It turns out that the Tits indices known from the classification of the semi-simple algebraic groups provide an interesting class of root gradings which are called stable. Any group with a stable root grading of rank 2 acts naturally on a bipartite graph which is called a Tits polygon. This action can be used to obtain classification results for groups with a stable root grading. I will report on several results in this direction. These have been obtained recently in joint work with Richard Weiss.