CAT(0) and acylindrical-hyperbolicity conjectures for Artin groups

Let Γ\Gamma be a finite simplicial graph, let AΓA_{\Gamma} be its Artin group, and write Z(AΓ)Z(A_{\Gamma}) for its centre. A group is CAT(0) here if it acts properly and cocompactly on a CAT(0) space.

Artin groups' geometric conjectures. For every Artin group AΓA_{\Gamma}: (1) AΓA_{\Gamma} is CAT(0); and (2) the central quotient AΓ/Z(AΓ)A_{\Gamma}/Z(A_{\Gamma}) is acylindrically hyperbolic.

These questions are known for right-angled Artin groups, some classes of two-dimensional Artin groups, spherical Artin groups of rank 33 for the CAT(0) assertion, and spherical Artin groups for the acylindrical-hyperbolicity assertion. They remain open in general.

Sources & referencesView supporting material

Primary source

Nicolas Vaskou, “Acylindrical hyperbolicity for Artin groups of dimension 2”, arXiv:2007.16169 (2021).

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