Séminaire de Mathématique
# Geometric construction of buildings for hyperbolic Kac-Moody groups

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Amphithéâtre Léon Motchane (IHES)
### Amphithéâtre Léon Motchane

#### IHES

Le Bois Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

A twin building is a simplicial complex associated to a group *G* with a twin BN-pair. A complex hyperbolic Kac-Moody (KM) group *G* is associated with a hyperbolic KM Lie algebra *g* = *g*(*A*), where *A* is a hyperbolic type Cartan matrix. The invariant symmetric bilinear form (. , .) on the standard Cartan subalgebra *h* in *g* has signature (*n*-1,1) on the split real form of *h*, providing a Lorentzian geometry. The Cartan-Chevalley involution on *g* gives a ``compact" real form *k* of *g*, a real Lie algebra whose complexification is *g*, whose Cartan subalgebra *t* also has Lorentzian form (. , .), and there is also a corresponding compact real group *K*. We are able to embed the twin building for *G* inside the union of all ``lightcones" {*x* in *k* | (*x*,*x*) <= 0} which is in the union of all *K* conjugates of *t*. This provides a geometrical realization of the twin building of *G* closely related to the structure of all Cartan subalgebras in *k*, and sheds light on the geometry of the infinite dimensional groups *G* and *K*. This is especially interesting in the case of rank 3 hyperbolic algebras whose Weyl groups are hyperbolic triangle groups, so that the building is a union of copies of the tesselated Poincaré disk with certain boundary lines identified. This is joint work with Lisa Carbone (Rutgers University) and Walter Freyn (TU-Darmstadt).

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