 
Summary: DUAL EUCLIDEAN ARTIN GROUPS AND THE
FAILURE OF THE LATTICE PROPERTY
JON MCCAMMOND
Abstract. The irreducible Coxeter groups that naturally act co
compactly on euclidean space are classified by the wellknown ex
tended Dynkin diagrams and these diagrams also encode the mod
ified presentations that define the irreducible Artin groups of eu
clidean type. These Artin groups remain mysterious with some
exceptions. Craig Squier clarified the structure of the three exam
ples with three generators more than twenty years ago and Fran¸cois
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 a Garside structure. This article
establishes that none of the remaining dual presentations for irre
ducible Artin groups of euclidean type are part of a Garside struc
ture because their factorization posets fail to be lattices. These are
the first examples of dual Artin presentations that are not Garside.
There is an irreducible Artin group of euclidean type for each of
the extended Dynkin diagrams. In particular, there are four infinite
families, An, Bn, Cn and Dn, known as the classical types plus five
