London Algebra Colloquium abstract

Discrete subgroups of locally finite Kac–Moody groups

Ralf Gramlich (Birmingham), 14th February 2008

Abstract

Let G be a split adjoint Kac–Moody group over a sufficiently large finite field of square order. The centraliser Γ in G of the twisted Chevalley involution is a natural generalisation of the concept of a unitary form of a finite Chevalley group. It turns out that Γ enjoys a number of quite remarkable algebraic and geometric properties, on which I would like to report:

• Γ is finitely presented, once the diagram of G is three-spherical (Devillers–Muehlherr plus G.–Hoffman–Muehlherr–Shpectorov),
• automorphisms of Γ can be uniquely extended to automorphisms of G (G.–Mars),
• Γ is of type F_n−1 but not of type F_n, where n is the rank of G (Devillers–G.–Muehlherr),
• Γ is a lattice in the completion of G with respect to the topology of uniform convergence on compact sets of the building G/B, where B is a Borel subgroup of G. Finally, if time allows I would like to point out recent developments in the theory of lattices in CAT(0) groups obtained by Caprace–Monod and discuss their concrete meaning for Γ.