The cohomology groups of the pure string motion group are uniformly representation stable

Geometry Topology Seminar
Monday, January 23, 2012 - 2:05pm for 1 hour (actually 50 minutes)
Skiles 006
Jenny Wilson – University of Chicago
Dan Margalit
In the past two years, Church, Farb and others have developed the concept of 'representation stability', an analogue of homological stability for a sequence of groups or spaces admitting group actions. In this talk, I will give an overview of this new theory, using the pure string motion group P\Sigma_n as a motivating example. The pure string motion group, which is closely related to the pure braid group, is not cohomologically stable in the classical sense -- for each k>0, the dimension of the H^k(P\Sigma_n, \Q) tends to infinity as n grows. The groups H^k(P\Sigma_n, \Q) are, however, representation stable with respect to a natural action of the hyperoctahedral group W_n, that is, in some precise sense, the description of the decomposition of the cohomology group into irreducible W_n-representations stabilizes for n>>k. I will outline a proof of this result, verifying a conjecture by Church and Farb.