On maximal dihedral reflection subgroups and generalized noncrossing partitions

In this note, we give a new proof of a result of Matthew Dyer stating that in an arbitrary Coxeter group W W, every pair t, t ′ t, t’ of distinct reflections lie in a unique maximal dihedral reflection subgroup of W W. Our proof only relies on the combinatorics of words, in particular we do not use root systems at all. As an application, we deduce a new proof of a recent result of Delucchi-Paolini-Salvetti, stating that the poset 1, c T 1, cT of generalized noncrossing partitions in any Coxeter group of rank 3 3 is a lattice. We achieve this by showing the more general statement that any interval of length 3 3 in the absolute order on an arbitrary Coxeter group is a lattice. This implies that the interval group attached to any interval 1, w T 1, wT where w w is an element of an arbitrary Coxeter group with ℓ T (w) = 3 T (w) =3 is a quasi-Garside group.