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What are
GT–shadows?
Vasily A Dolgushev, Khanh Q Le and Aidan A Lorenz
Algebraic & Geometric Topology 24 (2024) 2721–2777
Abstract
Let
B 4
(resp. PB 4 )
be the braid group (resp. the pure braid group) on
4 strands
and
N F I PB 4 ( B 4 )
be the poset whose elements are finite-index normal subgroups
N of
B 4 that are contained
in
PB 4 . We introduce
G T –shadows,
which may be thought of as “approximations” to elements of the profinite version
G T ^
of the Grothendieck–Teichmüller group (Drinfeld 1991). We prove that
G T –shadows
form a groupoid whose objects are elements of the underlying set
N F I PB 4 ( B 4 ) .
G T –shadows coming from
elements of
G T ^ satisfy
various additional properties and we investigate these properties. We establish an explicit link between
G T –shadows and the group
G T ^ . Selected results of computer
experiments on
G T –shadows
are presented. In the appendix we give a complete description of
G T –shadows
in the abelian setting. We also prove that, in the abelian setting, every
G T –shadow comes from an
element of
G T ^ . Objects
very similar to
G T –shadows
were introduced by D Harbater and L Schneps (1997). A variation of the concept of
G T –shadows for the
gentle version of
G T ^
was studied by P Guillot (2016 and 2018).
Keywords
Grothendieck–Teichmueller group, operads, braid groups, the
absolute Galois group of rational numbers
Mathematical Subject Classification
Primary: 14F35, 14H30, 18M60
Secondary: 14H57
Publication
Received: 25 July 2022
Revised: 8 April 2023
Accepted: 13 June 2023
Published: 19 August 2024
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