Mike Davis: Random right-angled Coxeter groups are rational duality groups

This is joint work with Matt Kahle. With various other coauthors, I have been computing the cohomology of a given graph product of groups with coefficients in, say, its group ring. The answers are given in terms of the ordinary cohomology groups of the flag complex associated to the graph.

Let G(n,p) be the "Erdos-Renyi random graph" on n vertices with edges inserted uniformly with probability p. In his thesis and some subsequent work, Kahle computed the cohomology of the flag complex associated to G(n, p) as n goes to infinity.

Combining Kahle's results with mine, we get computations of the cohomology of a random graph product of groups. One of the more striking results is that a random RACG (=right-angled Coxeter group) is a rational duality group. This means that it is type FP over the rationals and that its cohomology with coefficients in its group algebra is concentrated in a single degree which depends on (n, p). In fact, this degree turns out to be 1 + 1/2 the dimension of the associated flag complex.