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GT-shadows for the gentle version of the Grothendieck-Teichmueller group

Vasily A. Dolgushev, Jacob J. Guynee

Abstract

Let $B_3$ be the Artin braid group on 3 strands and $PB_3$ be the corresponding pure braid group. In this paper, we construct the groupoid $GTSh$ of GT-shadows for a (possibly more tractable) version $GT_0$ of the Grothendieck-Teichmueller group $GT$ introduced by D. Harbater and L. Schneps in 2000. We call this group the gentle version of $GT$ and denote it by $GT_{gen}$. The objects of $GTSh$ are finite index normal subgroups $N$ of $B_3$ satisfying the condition $N \subset PB_3$. Morphisms of $GTSh$ are called GT-shadows and they may be thought of as approximations to elements of $GT_{gen}$. We show how GT-shadows can be obtained from elements of $GT_{gen}$ and prove that $GT_{gen}$ is isomorphic to the limit of a certain functor defined in terms of the groupoid $GTSh$. Using this result, we get a criterion for identifying genuine GT-shadows.

GT-shadows for the gentle version of the Grothendieck-Teichmueller group

Abstract

Let be the Artin braid group on 3 strands and be the corresponding pure braid group. In this paper, we construct the groupoid of GT-shadows for a (possibly more tractable) version of the Grothendieck-Teichmueller group introduced by D. Harbater and L. Schneps in 2000. We call this group the gentle version of and denote it by . The objects of are finite index normal subgroups of satisfying the condition . Morphisms of are called GT-shadows and they may be thought of as approximations to elements of . We show how GT-shadows can be obtained from elements of and prove that is isomorphic to the limit of a certain functor defined in terms of the groupoid . Using this result, we get a criterion for identifying genuine GT-shadows.
Paper Structure (12 sections, 24 theorems, 258 equations)

This paper contains 12 sections, 24 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Notational conventions
  3. The gentle version $\widehat{\mathsf{GT}}_{gen}$ of the Grothendieck-Teichmueller group
  4. The monoid $( \widehat{\mathbb{Z}} \times {\widehat{\mathrm{F}}}_2, \bullet )$
  5. The monoid $\widehat{\mathsf{GT}}_{gen, mon}$ and the group $\widehat{\mathsf{GT}}_{gen}$
  6. The groupoid $\mathsf{GTSh}$
  7. The reduction map
  8. Connected compsonents of the groupoid $\mathsf{GTSh}$ and its isolated objects
  9. The transformation groupoid $\widehat{\mathsf{GT}}^{gen}_{\mathsf{NFI}}$ and genuine $\mathsf{GT}$-shadows
  10. The version of the Main Line functor for $\widehat{\mathsf{GT}}_{gen}$
  11. Simplified hexagon relations in the profinite setting
  12. Selected statements related to profinite groups

Key Result

Proposition 2.1

The set $\widehat{\mathbb{Z}} \times {\widehat{\mathrm{F}}}_2$ is a monoid with respect to the binary operation $\bullet$ (see bullet) and the pair $(0, 1_{ {\widehat{\mathrm{F}}}_2})$ is the identity element of this monoid. Moreover, the assignment defines a homomorphism of monoids $\widehat{\mathbb{Z}} \times {\widehat{\mathrm{F}}}_2 \to {\mathsf{End}}({\widehat{\mathrm{F}}}_2)$, where ${\maths

Theorems & Definitions (39)

  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 29 more