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Braids and the Grothendieck-Teichmuller Group
The "Grothendieck-Teichmuller Group" (GT) appears as a "depth certificate"
in many recent works - "we do A to B, apply the result to C, and get
something related to GT, therefore it must be interesting". Interesting
or not, in my talk I will explain how GT arose first, in Drinfel'd's work
on associators, and how it can be used to show that "every bounded-degree
associator extends", that "rational associators exist", and that "the
pentagon implies the hexagon".
In a nutshell: the filtered tower of braid groups (with bells and
whistles attached) is isomorphic to its associated graded, but the
isomorphism is neither canonical nor unique - such an isomorphism is
precisely the thing called "an associator". But the set of isomorphisms
between two isomorphic objects *always* has two groups acting simply
transitively on it - the group of automorphisms of the first object
acting on the right, and the group of automorphisms of the second object
acting on the left. In the case of associators, that first group is what
Drinfel'd calls the Grothendieck-Teichmuller group GT, and the second
group, isomorphic but not canonically to the first and denoted GRT,
is the one several recent works seem to refer to.
Almost everything I will talk about is in my old paper "On Associators and the
Grothendieck-Teichmuller Group I, also at arXiv:q-alg/9606021.
Handout. GT.html, GT.pdf,
GT.png (also as GT-1.png,
Sources. GT.zip, Pensieve: