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SUMMARY:The role of stability conditions in understanding Artin groups - E
 dmund Heng (IHES)
DTSTART:20221102T160000Z
DTEND:20221102T170000Z
UID:TALK184034@talks.cam.ac.uk
CONTACT:Oscar Randal-Williams
DESCRIPTION:The Artin groups (aka. Artin-Tits groups) are certain generali
 sations of the classical braid group. They can be realised as the fundamen
 tal groups of hyperplane complements\, or abstractly defined as lifts of C
 oxeter groups through generators and relations. Unlike Coxeter groups\, ho
 wever\, Artin groups are rather “mysterious”\, with lots of conjecture
 s remained unproven. \nRecently\, the theory of Bridgeland’s stability c
 onditions has been shown to have a strong potential in understanding the A
 rtin groups. Namely\, its striking connection with Teichmuller theory allo
 ws one to study actions of Artin groups on triangulated categories as repl
 acements for surfaces (it was known that certain Artin groups can not act 
 faithfully on surfaces). Moreover\, the space of stability conditions is e
 xpected 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 t
 ype Artin groups\, as there were no candidate category for the non-simply-
 laced ones to act on.\n\nIn this talk\, we shall complete the picture by f
 irst 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 condit
 ions\, which will be the main tool in obtaining the following results:\n\n
 1) A (categorical) Nielsen-Thurston classification for the rank two Artin 
 groups\; and\n2) The space of equivariant stability conditions as the K(\\
 pi\,1) space of the associated non-simply-laced\, finite type Artin group.
LOCATION:MR13
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