#### Volume 12, issue 2 (2012)

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Representation stability for the cohomology of the pure string motion groups

### Jennifer C H Wilson

Algebraic & Geometric Topology 12 (2012) 909–931
##### 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 ${H}^{k}\left(P{\Sigma }_{n};ℚ\right)$ are uniformly representation stable; that is, the description of the decomposition of ${H}^{k}\left(P{\Sigma }_{n};ℚ\right)$ into irreducible ${W}_{n}$–representations stabilizes for $n>>k$. We use a characterization of ${H}^{\ast }\left(P{\Sigma }_{n};ℚ\right)$ 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 ${H}^{k}\left({\Sigma }_{n};ℚ\right)$ 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.

##### Keywords
representation stability, homological stability, motion group, string motion group, circle-braid group, symmetric automorphism, basis-conjugating automorphism, braid-permutation group, hyperoctahedral group, signed permutation group
##### Mathematical Subject Classification 2000
Primary: 20J06, 20C15
Secondary: 20F28, 57M25