4:00 pm Monday, September 24, 2012
Colloquium: Random graph products of finite groups are rational duality groups by Make Davis (Ohio State U) in RLM 6.104
Let G(n,p) be the random graph on n vertices, i.e., G(n,p) is the probability space of graphs 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. Write f g to mean f=o(g). A famous result of Erdos-Renyi states that if p (log n)/n then asymptotically almost surely (abbreviated a.a.s.) X(n, p) is not connected, while if (log n)/n p it is connected a.a.s. 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. Submitted by
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