I have seen it conjectured several times that Artin groups are torsion-free. This is a very basic question one could ask about these groups. Intuitively, to me it seems like it must be true however it seems impossible to prove. I am curious if anyone is actually working on this / what kind of methods people may have tried to prove such a thing.



Torsion & Artin groups
This is a consequence of the [$K(\pi,1)$ conjecture](https://link.springer.com/chapter/10.1007%2F978-81-322-1814-2_13), stating that there is an explicit $K(\pi,1)$ for Artin groups which is the complement of a complex hyperplane arrangement whose real locus are the hyperplanes of the reflections in the associated Coxeter group quotient by the action of the Coxeter group. See Conjecture 2.2 in Paris' survey [Lectures on Artin groups and the $K(\pi, 1)$ conjecture](https://link.springer.com/chapter/10.1007%2F978-81-322-1814-2_13) for the statement and the discussion after for known cases. Since this is a finite-dimensional space (whose complex dimension is bounded by that of the number of generators), this implies that there cannot be any torsion. Recently Paolini and Salvetti have announced a [Proof of the $K(\pi, 1)$ conjecture for affine Artin groups](https://arxiv.org/abs/1907.11795). See also the introduction of their paper for a summary of previous results and McCammond's survey [The mysterious geometry of Artin groups](https://wbln.centre-mersenne.org/item/?id=WBLN_2017__4__A1_0): the affine case of the torsion-free problem was solved by McCammond and Sulway in [Artin groups of Euclidean type](https://link.springer.com/article/10.1007/s00222-017-0728-2).


