Sebastian Bischof - Groups of Kac-Moody type over F2

Abstract

Twin buildings are combinatorial objects which were introduced by Ronan and Tits in the late 1980s. Their definition was motivated by the theory of Kac-Moody groups over fields and they are natural generalizations of spherical buildings. Spherical buildings were classified by Tits in the 1970s and this result is based on a local-to-global result on spherical buildings. Tits asked the question whether a similar result holds for twin buildings. This has been confirmed by Mühlherr and Ronan under an additional assumption which is not satisfied for twin buildings associated with Kac-Moody groups over $F_2$. We give a construction of groups of Kac-Moody type over $F_2$ which shows that the local-to-global result does not hold in general. We will discuss some applications including finiteness properties and Property $T$.

Talk

Notation and historical context

We define a Coxeter system. There is a root (Building), they can be understood as half-spaces of the corresponding Cayley-Graph. A Prenilpotent pair of roots are given, when the intersection of roots and their negative roots is nonempty. In a building we have chambers and the Weyl distance.

A Twin building consists of two buildings, connected by the Coxeter system.

An RGD-System is a tuple consisting of a group and some Root group (Building). If the RGD-System is given over ${\mathbb{F}}_2$, then we have a special structure (see Structure theorem)

Let $\mathcal{P}$ be the set of prenilpotent positive root pairs. A Commutator blueprint of type $(W,S)$ is some family of ordered subsets.

For a commutator blueprint the three properties are introduced:

• Faithful commutator blueprint
• Weyl-invariant commutator blueprint
• Integrable commutator blueprint

We want to find sufficient condition for the above properties.

Commutator blueprints

Consequences

Keywords

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