|Date: Tuesday, January 10, 2012
Title: Random right-angled Coxeter groups are rational duality groups
Abstract: Erdos and Renyi developed the notion of a random graph on n vertices where each edge is inserted with uniform probability p. One then constructs X(n, p), the associated random flag complex (or clique complex) by filling in a simplex for each complete subgraph. The result of Erdos-Renyi is that if p is significantly larger than (log n)/n, then asymptotically almost surely (abbreviated a.a.s.) X(n, p) is connected. My collaborator, Matt Kahle, has generalized this by showing that in other regimes, the reduced cohomology of X(n, p) is a.a.s. concentrated in a single degree (with rational coefficients).
Associated to a graph and a group one can construct a new group called the "graph product." For example, when the group is cyclic of order 2, the graph product is a right-angled Coxeter group. One can compute the cohomology of such a graph product with coefficients in its group ring from the cohomology of flag complex associated to the graph. The notion of a random graph leads to the notion of a random graph product of groups. It follows that, with group ring coefficients, a random graph product of finite groups a.a.s. has cohomology concentrated in a single degree, i.e., it is a rational duality group.
Speaker: Mike Davis
Institution: Ohio State University