# Jennifer Wilson (University of Chicago)

## Monday 22 August, 12:05-12:55pm, Carslaw 375

### Representation stability for the cohomology of the groups of pure string motions

The string motion group $$\Sigma_n$$, the group of motions of $$n$$ disjoint, unlinked, unknotted circles in 3-space, is a generalization of the braid group. It can be identified with the symmetric automorphism group of the free group. The pure string motion group $$P\Sigma_n$$, the analogue of the pure braid group, admits an action by the hyperoctahedral group $$W_n$$. The rational cohomology of $$P\Sigma_n$$ is not stable in the classical sense -- the dimension of the $$k^{\mbox{th}}$$ cohomology group tends to infinity as $$n$$ grows -- however, Church and Farb have recently developed a notion of stability for a sequence of vector spaces with a group action, which they call representation stability. Inspired by their recent work on the cohomology of the pure braid group, they conjectured that for each $$k>0$$, the $$k^{\mbox{th}}$$ rational cohomology of $$P\Sigma_n$$ is uniformly representation stable with respect to the induced action of $$W_n$$, that is, the description of the decomposition of the cohomology group into irreducible $$W_n$$-representations stabilizes for $$n >> k$$. In this talk, I will give an overview of the theory of representation stability, and outline a proof verifying this conjecture. This result has implications for the cohomology of the string motion group, and the permutation-braid group.