Teichmüller discs in Schottkyspace

Actions of the Absolute Galois Group

We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{ü}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation came in part from his desire to understand the absolute Galois group. But he was also interested in Thurston's work on surfaces, and he expressed this in his Esquisse d'un programme, his R{é}coltes et semailles and on other occasions. He introduced the notions of dessin d'enfant…

On Grothendieck's tame topology

Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{ü}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surface topology and moduli spaces of Riemann surfaces, Grothendieck calls there for a recasting of topology, in order to make it fit to the objects of semialgebraic and semianalytic geometry, and in particular to the study of the Mumford-Deligne compactifications of moduli spaces. A …

A Motivic Grothendieck-Teichmüller Group

This paper proves the Beilinson-Soul{é} vanishing conjecture for motives attached to the moduli spaces of curves of genus 0 with n marked points. As part of the proof, it is also proved that these motives are mixed Tate. As a consequence of Levine's work, one obtains then well defined categories of mixed Tate motives over the moduli spaces of curves . It is shown that morphisms between moduli spaces forgetting marked points and embedding as boundary components induce functors between those categories and how tangential bases points fit in these functorialities. Tannakian formalism attaches gro…

直线与超平面 GT 变体

Line and Hyperplane GT-variants

本文利用超平面排列补集与 pro-ℓ 两步幂零基本群构造 GT 群的新变体，并证明其与有理数域的绝对 Galois 群同构。

The congruence subgroup property for the hyperelliptic modular group: the open surface case

Let ${\cal M}_{g,n}$ and ${\cal H}_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with $Γ_{g,n}$ and $H_{g,n}$, the so called Teichm{ü}ller modular group and hyperelliptic modular group. A choice of base point on ${\cal H}_{g,n}$ defines a monomorphism $H_{g,n}\hookrightarrowΓ_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichmüller group $Γ_{g,n}$ is the group of iso…