The K(π,1)K(\pi,1)-conjecture for infinite Coxeter groups

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Let WW be an infinite Coxeter group, let YWY_W be the associated hyperplane complement in C⊗V\mathbb{C}\otimes V restricted to the Tits cone, and let AA be the associated Artin group. For finite Coxeter groups, Deligne's theorem states that YW/WY_W/W is an aspherical space with fundamental group AA. The K(π,1)K(\pi,1)-conjecture. The analogue of Deligne's theorem holds for infinite Coxeter groups; in particular, YW/WY_W/W is aspherical and has fundamental group AA. Part of the conjecture, namely A=π1(YW/W)A=\pi_1(Y_W/W), is known for arbitrary Coxeter groups, and asphericity has been proved for some classes of infinite Coxeter groups, but the conjecture remains open for arbitrary Coxeter groups.

References

Primary source

Ruth Charney, “An introduction to right-angled Artin groups”, arXiv:math/0610668 (2006).

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