Extended weak order in all rank three Coxeter groups

Abstract

We prove that extended weak order is a lattice for all rank 3 Coxeter groups. This was previously known in finite and affine types, and in the universal Coxeter group of rank 3. All other cases are new to this paper. As in earlier work, the proof proceeds by showing that all rank 3 Coxeter arrangements are clean arrangements, meaning that every biclosed set is weakly separated from its complement by a hyperplane. As a consequence, we also deduce Dyer's Conjecture A for all rank 3 Coxeter groups, which asserts that cover relations in extended weak order are given by toggling a single root.