On maximal dihedral reflection subgroups and generalized noncrossing partitions (Q6602142)

Let \((W, S)\) be a Coxeter system. It is a theorem of Matthew Dyer that any two distinct reflections in \(W\) are contained in a unique maximal dihedral reflection subgroup. In the paper under review, the author provides a purely combinatorial proof of this result without relying on the realized root system in the vector space. Along the way, some properties of the maximal dihedral reflection subgroups are also established using only the combinatorics of words.\N\NIn the second part of the paper, the author uses these properties and the aforementioned Dyer's theorem to show that any interval of length 3 in the absolute order on \(W\) is a lattice. Two interesting consequences of this result are then given. The first is a new combinatorial proof of a theorem of Delucchi, Paolini, and Salvetti in [\textit{E. Delucchi} et al., Geom. Topol. 28, No. 9, 4295--4336 (2024; Zbl 07961587)], which states that for any Coxeter system \((W, S)\) of rank \(3\) and a Coxeter element \(c \in W\), the interval \([1, c]\) in the absolute order is a lattice. The second consequence is that for any element \(w\in W\) with absolute length 3, the interval group attached to the interval \([1, w]\) (in the absolute order) is a quasi-Garside group.