# Basic Questions on Artin-Tits groups

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 $\infty$.
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https://hal.archives-ouvertes.fr/hal-00590858
Contributor : Eddy Godelle <>
Submitted on : Thursday, May 5, 2011 - 12:01:41 PM
Last modification on : Thursday, February 7, 2019 - 5:42:27 PM
Long-term archiving on : Saturday, August 6, 2011 - 2:47:06 AM

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### Identifiers

• HAL Id : hal-00590858, version 1
• ARXIV : 1105.1048

### Citation

Eddy Godelle, Luis Paris. Basic Questions on Artin-Tits groups. 2011. ⟨hal-00590858⟩

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