Right-angled Artin subgroup of Artin groups

30 November 2020
Kasia Jankiewicz

Artin groups are a family of groups generalizing braid groups. The Tits conjecture, which was proved by Crisp-Paris, states that squares of the standard generators generate an obvious right-angled Artin subgroup. In a joint work with Kevin Schreve, we consider a larger collection of elements, and conjecture that their sufficiently large powers generate an obvious right-angled Artin subgroup. In the case of the braid group, regarded as a mapping class group of a punctured disc, these elements correspond to Dehn twist around the loops enclosing multiple consecutive punctures. This alleged right-angled Artin group is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for some classes of Artin groups. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid.

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