Representation stability for the cohomology of the pure string motion groups

TL;DR

This paper proves that the rational cohomology groups of the pure string motion group exhibit uniform representation stability and vanish in positive degrees, with implications for the cohomology of related groups.

Contribution

It confirms Church and Farb's conjecture on representation stability for the pure string motion group's cohomology and establishes stability for the braid-permutation subgroup.

Findings

Cohomology groups stabilize in irreducible W_n representations for large n.

Rational cohomology of the string motion group vanishes in positive degrees.

Braid-permutation group is rationally cohomologically stable.

Abstract

The cohomology of the pure string motion group PSigma_n admits a natural action by the hyperoctahedral group W_n. Church and Farb conjectured that for each k > 0, the sequence of degree k rational cohomology groups of PSigma_n is uniformly representation stable with respect to the induced action by W_n, that is, the description of the groups' decompositions into irreducible W_n representations stabilizes for n >> k. We use a characterization of the cohomology groups 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 in positive degree. We also prove that the subgroup of orientation-preserving string motions, also known as the braid-permutation group, is rationally cohomologically stable in the classical sense.