  This paper is a short survey on four basic questions on Artin-Tits groups:
the torsion, the center, the word problem, and the cohomology ($K(\pi,1)$
problem). It is also an opportunity to prove three new results concerning these
questions: (1) if all free of infinity Artin-Tits groups are torsion free, then
all Artin-Tits groups will be torsion free; (2) If all free of infinity
irreducible non-spherical type Artin-Tits groups have a trivial center then all
irreducible non-spherical type Artin-Tits groups will have a trivial center;
(3) if all free of infinity Artin-Tits groups have solutions to the word
problem, then all Artin-Tits groups will have solutions to the word problem.
Recall that an Artin-Tits group is free of infinity if its Coxeter graph has no
edge labeled by $\infty$.
