On Associators and the Grothendieck-teichmuller Group I

On Associators and the Grothendieck - Teichmuller

We present a formalism within which the relationship (discovered by Drinfel’d in [Dr1, Dr2]) between associators (for quasi-triangular quasi-Hopf algebras) and (a variant of) the Grothendieck-Teichmuller group becomes simple and natural, leading to a great simplification of Drinfel’d’s original work. In particular, we re-prove that rational associators exist and can be constructed iteratively.

On Associators and the Grothendieck - Teichmullergroup

Drinfel’d-Ihara Relations for the Crystalline Frobenius

For K a field of characteristic zero, let M(K) denote the set of Drinfel’d associators defined over K [Dr]. The variety M is of great interest because of its connection to the deformations of universal enveloping algebras, fundamental group of the Teichmuller tower, and to Gal(Q/Q). There is a natural map M → A and for λ ∈ K, and Mλ is a torsor under GRT1 (= M0). If K << X,Y >> denotes the form...

Arithmetic Teichmuller Theory

By Grothedieck's Anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number fields encode all arithmetic information of these curves. The goal of this paper is to develope and arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing th...

The Weil-Petersson Isometry Group

Let F = Fg,n be a surface of genus g with n punctures. We assume 3g − 3 + n > 1 and that (g, n) 6= (1, 2). The purpose of this paper is to prove, for the Weil-Petersson metric on Teichmuller space Tg,n, the analogue of Royden’s famous result [15] that every complex analytic isometry of Tg,0 with respect to the Teichmuller metric is induced by an element of the mapping class group. His proof inv...