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Abstract

Artin—Tits groups are fundamental groups of complexified hyperplane complements. Examples of these groups include the braid groups, which can be viewed as mapping class groups of punctured disks. In these cases, Teichmuller theory offers extremely powerful tools, making braid groups one of the most well-understood family among the Artin—Tits groups. While one might hope to study Artin—Tits groups via mapping class groups in a similar fashion, Wajynrib has proven a technical result that makes it hard for Artin—Tits groups (in particular, type E) to act faithfully on surfaces.

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The aim of this talk is to convince you of an a priori different approach that is nevertheless highly motivated by the theory of surfaces — we study Artin—Tits groups via actions on triangulated categories. More precisely, we will build on an analogy between surfaces and triangulated category initiated by Dmitrov, Haiden, Katzarkov and Kontsevich, where the theory of Bridgeland stability conditions plays the role of Teichmuller theory.

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In the first talk, I will introduce the simplest example of a triangulated category: the homotopy category of complexes over nice additive categories. In fact, all triangulated categories of interests in both talks are of this form. I will introduce an important tool called “Gaussian elimination” (coined by Bar-Natan), which allows one to simplify any complex into a minimal form up to homotopy. In our setting, this simple tool already offers a solution to the word problem for spherical type Artin—Tits groups, which moreover conjecturally works for all types.

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In the second talk, I will discuss a recent progress on the categorical dynamics of spherical Artin—Tits groups. In particular, I will speak about how periodic elements have fixed points on the space of Bridgeland stability conditions — similar to how periodic elements have fixed points on the Teichmuller space, in the sense of the dynamical classification of Nielsen and Thurston. This is joint work with Oded Yacobi and Tony Licata.