Recovering the Artin–Tits group from the PL structure

Determine to what extent the Artin–Tits group of a Coxeter system can be recovered from its piecewise-linear action on the simplicial complex \(\Sigma\) associated with a generic stability condition.

Background

The paper proves in type A that the braid group acts on the simplicial complex by piecewise-linear homeomorphisms and expects analogous behavior for Artin–Tits groups associated with general Coxeter systems. Motivated by rigidity phenomena for mapping class groups acting on spaces of projective measured foliations, it asks whether the group can be reconstructed from the PL structure and its preserved functions.