The cohomology of the pure string motion group admits a natural action by the hyperoctahedral group . In recent work, Church and Farb conjectured that for each , the cohomology groups are uniformly representation stable; that is, the description of the decomposition of into irreducible –representations stabilizes for . We use a characterization of 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 vanish for . We also prove that the subgroup of of orientation-preserving string motions, also known as the braid-permutation group, is rationally cohomologically stable in the classical sense.
"Representation stability for the cohomology of the pure string motion groups." Algebr. Geom. Topol. 12 (2) 909 - 931, 2012. https://doi.org/10.2140/agt.2012.12.909