Basic Questions on Artin-Tits groups

TL;DR

This paper surveys key questions about Artin-Tits groups, including torsion, center, word problem, and cohomology, and proves new results linking properties of free of infinity groups to all Artin-Tits groups.

Contribution

It establishes that properties like torsion-freeness, trivial center, and solvability of the word problem in free of infinity Artin-Tits groups imply these properties for all Artin-Tits groups.

Findings

If all free of infinity Artin-Tits groups are torsion free, then all Artin-Tits groups are torsion free.

If all free of infinity irreducible non-spherical Artin-Tits groups have trivial center, then all such groups have trivial center.

If all free of infinity Artin-Tits groups have solutions to the word problem, then all Artin-Tits groups do.

Abstract

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 $\infty$.