Dual Euclidean Artin Groups and the Failure of the Lattice Property

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 corrspond to Garside structures because their factorization posets fail to be lattices. These are the first known examples of dual Artin presentations that 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. There is an irreducible Artin group of euclidean type for each of the extended Dynkin diagrams. In particular, there are four infinite families, Ãn, B̃n, C̃n and D̃n, known as the classical types plus five remaining exceptional examples Ẽ8, Ẽ7, Ẽ6, F̃4, and G̃2. Most of these groups have been poorly understood until very recently. Among the few known results are a clarification of the structure of the Artin groups of types Ã2, C̃2 and G̃2 by Craig Squier in [Squ87] and two papers by François Digne [Dig06, Dig12] proving that the Artin groups of type Ãn and C̃n have dual presentations that are Garside structures. Our main result is that Squier’s and Digne’s examples are the only ones that have dual presentations that are Garside structures. Date: November 12, 2014. 2010 Mathematics Subject Classification. 20F36.