◆ PRO-ALGEBRAIC BRAID GROUP

IHARA BRAID ACTION // OUTER AUTOMORPHISMS // GROTHENDIECK-TEICHMULLER // PRO-UNIPOTENT GT

I. IHARA'S DISCOVERY

Ihara discovered that Gal(Q/Q) acts on the profinite free group F_2^ = pi_1^et(P^1\{0,1,inf}) via outer automorphisms. Each sigma determines (chi(sigma), f(sigma)) where chi is cyclotomic and f lives in the commutator subgroup.

Gal(Q/Q) -> Out(F_2^) injective (Belyi)

II. GROTHENDIECK-TEICHMULLER

GT^ is defined by pentagon, hexagon, and inversion relations in the profinite free group. Drinfeld proved Gal(Q/Q) embeds into GT^. The conjecture Gal(Q/Q) = GT^ would reduce number theory to coherent deformations of associativity.

GT^ = { (lambda, f) : pentagon + hexagon + inversion }

III. THE TOWER OF M_{0,n}

M_{0,n+1} -> M_{0,n} forms a tower whose pro-unipotent fundamental groups carry an operad structure. GT acts on the entire tower compatibly. At the bottom, M_{0,4} = P^1\{0,1,inf}. The KZ associator is a special element of GT_1.

M_{0,n+1} -> M_{0,n} -> ... -> M_{0,4} = P^1\{0,1,inf}

IV. PRO-UNIPOTENT GT

The graded Lie algebra grt_1 is generated by sigma_{2n+1} in degree 2n+1. These correspond to odd zeta values. The Drinfeld-Ihara Lie algebra captures the pro-unipotent part of the Galois action.

grt_1 = Lie(sigma_3, sigma_5, sigma_7, ...)

V. BRAID GROUPS

Pure braid groups PB_n have pro-unipotent completions whose Lie algebras are Drinfeld-Kohno. They arise in deformation quantization and Kontsevich formality.

PB_n^un ~ exp(t_{ij} / infinitesimal braid rels)
C445 // PRO-ALGEBRAIC BRAID // GROTHENDIECK ANABELIA