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The Betti side of the double shuffle theory. III. Bitorsor structures

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Pith's one-line read An explicit bitorsor structure is constructed on the double shuffle torsor, with right-acting Betti group DMR_B(k) whose discrete analogue is {±1} and whose pro-p version fits a Cartesian diagram with GT_p.

arxiv 1908.00444 v5 pith:R2VKNF4X submitted 2019-08-01 math.AG math.NTmath.QA

Benjamin Enriquez , Hidekazu Furusho This is my paper · ORCID
classification math.AGmath.NTmath.QA
keywords bettibitorsordoubleshufflenatureseriesstructurestorsors
verification ladder T0 review T1 audit T2 compute T3 formal

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The reading

Algebraic geometry and number theory often study 'torsors': sets equipped with a group that can move any element to any other, like a clock face under rotations. A 'bitorsor' adds a second group acting from the right. The associators of Drinfeld and the solutions of the double shuffle equations form torsors; this paper makes the missing right actions explicit. The authors construct a new group, DMR_B(k), that acts on the right of the double shuffle torsor, and show it contains the Grothendieck-Teichmüller group from the left side. This 'Betti' version is built from the free group on two generators and a coproduct called the harmonic coproduct, rather than from the de Rham differential forms used in earlier parts. A discrete analogue, obtained by intersecting with {±1} ⋉ F2, is computed to be exactly {±1}: the only discrete symmetries are trivial up to sign. The paper also defines a pro-p analogue DMR_B^p as the intersection of DMR_B(Q_p) with Z_p^× ⋉ F_2^(p), and proves it fits into a Cartesian diagram with the pro-p Grothendieck-Teichmüller group GT_p. The upshot is a cleaner algebraic picture of how associators and double shuffle solutions relate.
Extended reading notes

Core claim

The load-bearing assertion is Theorem 3.14(a): the group attached to the subtorsor Stab(∆_{W,DR/B})(k) (resp. Stab(∆_{M,DR/B})(k), DMR_DR,B(k), DMR_μ(k)) of the torsor G_DR,B(k) is the subgroup Stab(∆_{W,B})(k) (resp. Stab(∆_{M,B})(k), DMRB(k), DMRB_0(k)) of GB(k). This makes the bitorsor structures explicit, defines DMRB(k) as the Betti counterpart of the Grothendieck-Teichmüller group, and supports the derived discrete computation DMRB = {±1} (Proposition 5.10) and the pro-p analogue DMRB_p (Definition 6.16).

Load-bearing premise

The paper depends on the authors' earlier equality DMR_DR,B(k) = Stab(∆_{M,DR/B})(k) ∩ G_DR,B_quad(k) and the inclusion M(k) ⊂ DMR_DR,B(k), imported from [EF2, Theorem 3.1] and used in the proofs of Lemma 3.11 and Theorem 3.14(a). This equality is what turns the new stabilizer subgroups into a bitorsor structure on the double shuffle torsor; if it were false, the identification of DMRB(k) as the right-acting group would not follow. The paper also asserts in Section 3.8, without proof, that the functors are Q-group schemes, which the pro-p section treats as Q_p-points of such schemes.

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Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper is a sequel and explicitly imports structural results from [EF1], [EF2], [D], and [R]. No free parameters are fitted. The main constructed object, DMRB(k), is defined from known ingredients (a stabilizer of the harmonic coproduct) and is then tested by the discrete computation and the pro-p Cartesian diagram, which serve as internal independent checks.

assumptions (5)
  • standard math Torsor-bitorsor equivalence: every torsor GX carries a canonical right action of Aut_G(X), giving a bitorsor GXAut_G(X).
    Used throughout Section 3 to attach right-acting groups; cited to Giraud, Proposition 1.5.1 (Lemma 1.11).
  • domain assumption The Betti and de Rham data are related by Hopf isomorphisms iso_V: V-hat_B → V-hat_DR, iso_W, iso_M from [EF1], §3.3.
    Lemma 2.10 and 2.14 rely on these isomorphisms to transfer group structures and define the twisted Magnus bitorsor; the construction is in the authors' earlier paper [EF1].
  • domain assumption DMR_DR,B(k) = Stab(∆_{M,DR/B})(k) ∩ G_DR,B_quad(k) and M(k) ⊂ DMR_DR,B(k) (from [EF2], Theorem 3.1).
    This is the bridge used in Lemma 3.11 and Theorem 3.14(a) to identify DMRB(k) as the right-acting group; if this equality failed, the bitorsor identification for the double shuffle torsor would collapse.
  • domain assumption Non-emptyness of associator set M1(Q) and Drinfeld's Lie algebra isomorphisms b_Φ^P: Lie(K_n) ≃ t_n-hat for associators Φ (Drinfeld [D]).
    Used in the proof of Theorem 4.5 (to produce (1,φ) ∈ DMR_DR,B(k)) and in Lemma 6.7 (exactness of prounipotent completions of braid groups).
  • domain assumption Exactness of pro-p completion sequences 1 → N^(p) → G^(p) → H^(p) → 1 for free normal subgroups (Ihara, Anderson; Lemma 6.5) and the identification Z_p[[F_n^(p)]] ≃ A(n) (Serre; Lemma 6.3).
    Used in Section 6 to prove injectivity K_4^(p) → K_4(Q_p), which underlies Propositions 6.12 and 6.18.
invented entities (1)
  • DMRB(k), the Betti double shuffle group scheme independent evidence
    purpose: Acts freely and transitively on the right of the double shuffle torsor DMR_DR,B(k); Betti counterpart of GT(k).
    Defined as GB_quad(k) ∩ Stab(∆_{M,B})(k) (Lemma-Definition 3.10), so it is constructed rather than postulated. Independent evidence: discrete analogue DMRB = {±1} (Proposition 5.10) and the Cartesian diagram for the pro-p version DMRB_p (Proposition 6.18).

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Pith. "Pith review of The Betti side of the double shuffle theory. III. Bitorsor structures." pith.science (2026). https://pith.science/paper/R2VKNF4X

@misc{pith2026190800444,
  author       = {Pith},
  title        = {Pith review of: The Betti side of the double shuffle theory. III. Bitorsor structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R2VKNF4X}},
  note         = {Machine review of arXiv:1908.00444}
}
read the original abstract

In the two first parts of the series, we constructed stabilizer subtorsors of a `twisted Magnus' torsor, studied their relations with the associator and double shuffle torsors, and explained their `de Rham' nature. In this paper, we make the associated bitorsor structures explicit and explain the `Betti' nature of the corresponding right torsors; we thereby complete one aim of the series. We study the discrete and pro-p versions of the `Betti' group of the double shuffle bitorsor.

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Works this paper leans on

14 extracted references · 14 canonical work pages

  1. [1]

    Anderson, Exactness properties of pro-finite functors , Topology 13 (1974), 229--239

    M.P. Anderson, Exactness properties of pro-finite functors , Topology 13 (1974), 229--239

    work page 1974

  2. [2]

    Andr\'e, Une introduction aux motifs (motifs purs, motifs mixtes, p\'eriodes)

    Y. Andr\'e, Une introduction aux motifs (motifs purs, motifs mixtes, p\'eriodes). Panoramas et Synth\`eses, 17. Soci\'et\'e Math\'ematique de France, Paris, 2004

    work page 2004

  3. [3]

    Bourbaki, \'El\'ements de math\'ematique

    N. Bourbaki, \'El\'ements de math\'ematique. Fasc. XXXVII. Groupes et alg\`ebres de Lie. Chapitre II: Alg\`ebres de Lie libres. Chapitre III: Groupes de Lie, Actualit\'es Scientifiques et Industrielles, No. 1349. Hermann, Paris, 1972

    work page 1972

  4. [4]

    Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal ( Q / Q ) , Leningrad Math

    V. Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal ( Q / Q ) , Leningrad Math. J. 2 (1991), no. 4, 829--860

    work page 1991

  5. [5]

    Deligne, A

    P. Deligne, A. Goncharov, Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), no. 1, 1--56

    work page 2005

  6. [6]

    Enriquez, H

    B. Enriquez, H. Furusho, A stabilizer interpretation of double shuffle Lie algebras. Int. Math. Res. Not. IMRN 2018, no. 22, 6870--6907

    work page 2018

  7. [7]

    The Betti side of the double shuffle theory. I. The harmonic coproducts

    B. Enriquez, H. Furusho, The Betti side of the double shuffle theory. I. The harmonic coproduct. Preprint arXiv:1803.10151, v5, to appear in Selecta Math

    work page Pith review arXiv

  8. [8]

    The Betti side of the double shuffle theory. II. Double shuffle relations for associators

    B. Enriquez, H. Furusho, The Betti side of the double shuffle theory. II. Double shuffle relations for associators. Preprint arXiv:1807.07786, v3

    work page Pith review arXiv

Show all 14 references
  1. [9]

    Furusho, Double shuffle relation for associators

    H. Furusho, Double shuffle relation for associators. Ann. of Math. (2) 174 (2011), no. 1, 341--360

    2011

  2. [10]

    Giraud, Cohomologie non ab\' e lienne

    J. Giraud, Cohomologie non ab\' e lienne. Die Grundlehren der mathematischen Wissenschaften, Band 179. Springer-Verlag, Berlin-New York, 1971

    1971

  3. [11]

    R. Hain, M. Matsumoto, Weighted completion of Galois groups and Galois actions on the fundamental group of P ^1-\ 0,1, \ , Compositio Math. 139 (2003), no. 2, 119--167

    2003

  4. [12]

    Ihara, Automorphisms of pure sphere braid groups and Galois representations , The Grothendieck Festschrift, Vol

    Y. Ihara, Automorphisms of pure sphere braid groups and Galois representations , The Grothendieck Festschrift, Vol. II, 353--373, Progr. Math., 87, Birkh\"auser Boston, Boston, MA, 1990

    1990

  5. [13]

    Racinet, Doubles m\'elanges des polylogarithmes multiples aux racines de l'unit\'e

    G. Racinet, Doubles m\'elanges des polylogarithmes multiples aux racines de l'unit\'e. Publ. Math. Inst. Hautes \'Etudes Sci., No. 95 (2002), 185--231

    2002

  6. [14]

    Serre, Cohomologie galoisienne

    J.-P. Serre, Cohomologie galoisienne. Fifth edition. Lecture Notes in Mathematics, 5. Springer-Verlag, Berlin, 1994

    1994

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