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ARTIN GROUPS OF EUCLIDEAN TYPE JON MCCAMMOND AND ROBERT SULWAY
 

Summary: ARTIN GROUPS OF EUCLIDEAN TYPE
JON MCCAMMOND AND ROBERT SULWAY
Abstract. This article resolves several long-standing conjectures
about Artin groups of euclidean type. We prove, in particular, that
every irreducible euclidean Artin group is a torsion-free centerless
group with a decidable word problem and a finite-dimensional clas-
sifying space. We do this by showing that it is isomorphic to a sub-
group of a Garside group in the expanded sense of Digne. The Gar-
side groups involved are introduced here for the first time. They
are constructed by applying semi-standard procedures to crystal-
lographic groups that contain euclidean Coxeter groups but which
need not be generated by the reflections they contain.
Arbitrary Coxeter groups are groups defined by a particularly simple
type of presentation, but the central motivating examples that lead to
the general theory are the groups generated by reflections that act
properly discontinuously and cocompactly by isometries on spheres
and euclidean spaces. Presentations for these spherical and euclidean
Coxeter groups are encoded in the well-known Dynkin diagrams and
extended Dynkin diagrams, respectively. Arbitrary Artin groups are
groups defined by a modified version of these simple presentations, a

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics