An Artin group of finite rank is a group defined by a Coxeter matrix 
on a finite set 
. It has generators 
for 
. For distinct 
with 
, its defining relation
equates the alternating words 
and 
of length 
.
The value 
imposes no relation between that pair. For 
the generators commute, while 
gives 
. Adding the
relations 
gives the associated Coxeter group.
For example, a graph path of 
generators, with label 3 between adjacent generators and
label 2 between other pairs, defines the braid group 
. The standard topological model is the Artin
Salvetti complex, whose asphericity is the
Artin K(pi,1) conjecture.
A metric space is proper when every closed bounded subset is compact.
Conjecture for Artin Groups." Ann. Fac. Sci.
Toulouse Math. 23, 361-415, 2014.