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.
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4$\begingroup$ I know S. Rees and D. Holt have worked on this and related areas not too long ago. I think these slides could be a good place to start. D. Holt is very active on here, so he can probably give a good answer to this, if anyone can! It seems to me that a large chunk of the research is focussed on solving the word problem for Artin groups (which remains open afaik), but I have not looked at this very extensively myself. $\endgroup$– Carl-Fredrik Nyberg BroddaCommented Aug 2, 2020 at 21:03
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$\begingroup$ Thanks for the reply! I'll take a look $\endgroup$– Dominic PettiCommented Aug 2, 2020 at 21:14
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2$\begingroup$ All of the results about Artin groups that I know of concern specific types of Artin groups (spherical type, large type, right angled, and variations of these). Very little is known about Artin groups in general - for example, it is not known whether the word problem is solvable. One example of a general result (due to Crisp and Paris) is that the subgroup generated by the squares of the generators is a right angled Artin group with $[x^2,y^2] = 1 \Leftrightarrow [x,y]=1$. $\endgroup$– Derek HoltCommented Aug 2, 2020 at 21:59
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1$\begingroup$ @DerekHolt's reference: Crisp and Paris - The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group (MSN). $\endgroup$– LSpiceCommented Aug 2, 2020 at 23:28
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$\begingroup$ @DerekHolt oh cool! I’ll take a look $\endgroup$– Dominic PettiCommented Aug 2, 2020 at 23:33
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1 Answer
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This is a consequence of the $K(\pi,1)$ conjecture, 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 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. See also the introduction of their paper for a summary of previous results and McCammond's survey The mysterious geometry of Artin groups: the affine case of the torsion-free problem was solved by McCammond and Sulway in Artin groups of Euclidean type.
