Dual euclidean Artin groups and the failure of the lattice property

Abstract

The irreducible euclidean Coxeter groups that naturally act geometrically on euclidean space are classified by the well-known extended Dynkin diagrams and these diagrams also encode the modified presentations that define the irreducible euclidean Artin groups. These Artin groups have remained mysterious with some exceptions until very recently. Craig Squier clarified the structure of the three examples with three generators more than twenty years ago and François Digne more recently proved that two of the infinite families can be understood by constructing a dual presentation for each of these groups and showing that it forms an infinite-type Garside structure. In this article I establish that none of the remaining dual presentations for irreducible euclidean Artin groups correspond to Garside structures because their factorization posets fail to be lattices. These are the first known examples of Artin groups where all of their dual presentations fail to form Garside structures. Nevertheless, the results presented here about the cause of this failure form the foundation for a subsequent article in which the structure of euclidean Artin groups is finally clarified.