The curvature of posets associated to the braid groups
Abstract: There is a standard construction in enumerative combinatorics for turning a finite graded poset into a simplicial complex called its order complex. In this talk I will describe work in progress with Tom Brady (Dublin City University) which combines this technique with the concept of non-positive curvature from geometric group theory. In particular, we assign a piecewise spherical metric to the order complex of a poset and ask whether the result is a CAT(1) metric space.
The particular example we are most interested in is the lattice of noncrossing partitions. If the metric on the order complex of the lattice of noncrossing partitions is CAT(1), then the braid groups are non-positively curved.