Mathematics Colloquia and Seminars

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Buildings are special simplicial complexes associated to Coxeter groups, and hence Lie groups. While spherical buildings have simplices which correspond to subgroups whose quotient is compact, Euclidean buildings are (in certain nice cases) the reverse: simplices correspond to precompact subgroups. Geometrically the biggest difference is that spherical buildings act a lot like positively curved manifolds, while Euclidean buildings act like non-positively curved manifolds (although both are highly singular). To make this rigorous, we define a generalization of being non-positively curved to metric spaces. This is called the CAT(0) property. Similar to the hyperbolic case, we can then define a space at infinite whose points are equivalence classes of rays, and perhaps surprisingly it turns out that this space is a spherical building, thus giving a direct connection between these two types of buildings. Knowledge of my previous talk will not be assumed.