Representation stability for the cohomology of the pure string motion groups

Abstract

The cohomology of the pure string motion group $P{\Sigma }_n$ admits a natural action by the hyperoctahedral group $W_n$. In recent work, Church and Farb conjectured that for each $k\ge 1$, the cohomology groups $H^k\left(P{\Sigma }_n;\mathbb{Q}\right)$ are uniformly representation stable; that is, the description of the decomposition of $H^k\left(P{\Sigma }_n;\mathbb{Q}\right)$ into irreducible $W_n$–representations stabilizes for $n>>k$. We use a characterization of $H^{\ast }\left(P{\Sigma }_n;\mathbb{Q}\right)$ given by Jensen, McCammond and Meier to prove this conjecture. Using a transfer argument, we further deduce that the rational cohomology groups of the string motion group $H^k\left({\Sigma }_n;\mathbb{Q}\right)$ vanish for $k\ge 1$. We also prove that the subgroup of ${\Sigma }_n^{+}\subseteq {\Sigma }_n$ of orientation-preserving string motions, also known as the braid-permutation group, is rationally cohomologically stable in the classical sense.