[Binghamton Geometry/Topology Seminar] Geometry & Topology seminar/ Dean's Speaker

[somnath at math.binghamton.edu]

BINGHAMTON GEOMETRY/TOPOLOGY SEMINAR
(DEAN's SPEAKER SERIES)

*Date:*  Wednesday, Dec 5, 2012

*Time:*  3:30-4:30pm
*Place:*  Library North 2205 followed by coffee/tea in the Anderson Reading
Room.

*Speaker: ** *Mike Davis (Ohio State University)
*Title: *Random graph products of finite groups are rational duality groups
 *
Abstract:* Let G(n,p) be the Bernoulli random graph on n vertices, i.e.,
G(n,p) is the probability space of all graphs on n vertices where each
edge is inserted with uniform probability p. Let X(n, p) be the associated
random flag complex (or clique complex) obtained by filling in a simplex
for each complete subgraph. Usually p will be a function of n. Write f <<
g to mean f = o(g). A famous result of Erdos-Renyi states that if p <<
(log n)/n, then, with high probability (abbreviated w.h.p.), X(n, p) is
not connected, while if (log n)/n << p, it is connected w.h.p. My
collaborator, Matt Kahle, has generalized this by showing that in other
ranges of p, the reduced cohomology (with rational coefficients) of X(n,
p) is concentrated in a single degree w.h.p. It turns out that the degree
where the cohomology is concentrated is the greatest integer < d/2, where
d is the dimension of X(n,p). One can also associate to a graph and a
sequence of groups indexed by the vertex set of the graph, a new group
called the "graph product." For example, when each 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 of groups, with
coefficients in its group algebra, from the cohomology of the flag complex
determined by the graph. So, the notion of a random graph leads to the
notion of a random graph product of groups. In previous work I calculated
the cohomology of graph products in terms of the cohomology of the
associated flag complex. For example, it follows that, with group algebra
coefficients, a random graph product of finite groups .w.h.p. has
cohomology concentrated in a single degree, i.e., is a rational duality
group.