The role of stability conditions in understanding Artin groups

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• Edmund Heng (IHES)
• Wednesday 02 November 2022, 16:00-17:00
• MR13.

If you have a question about this talk, please contact Oscar Randal-Williams.

The Artin groups (aka. Artin-Tits groups) are certain generalisations of the classical braid group. They can be realised as the fundamental groups of hyperplane complements, or abstractly defined as lifts of Coxeter groups through generators and relations. Unlike Coxeter groups, however, Artin groups are rather “mysterious”, with lots of conjectures remained unproven. Recently, the theory of Bridgeland’s stability conditions has been shown to have a strong potential in understanding the Artin groups. Namely, its striking connection with Teichmuller theory allows one to study actions of Artin groups on triangulated categories as replacements for surfaces (it was known that certain Artin groups can not act faithfully on surfaces). Moreover, the space of stability conditions is expected to be the K(\pi,1) space for the associated Artin group. However, most of the results (or expected results) only allow for simply-laced type Artin groups, as there were no candidate category for the non-simply-laced ones to act on.

In this talk, we shall complete the picture by first introducing certain triangulated categories (equipped with actions of fusion categories) that the non-simply-laced Artin groups act on. Then, we shall introduce the notion of “fusion-equivariant” stability conditions, which will be the main tool in obtaining the following results:

1) A (categorical) Nielsen-Thurston classification for the rank two Artin groups; and 2) The space of equivariant stability conditions as the K(\pi,1) space of the associated non-simply-laced, finite type Artin group.

This talk is part of the Differential Geometry and Topology Seminar series.

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