Relative Pro-$\ell$ Completions of Mapping Class Groups

TL;DR

This paper develops the theory of relative pro-ell completions of groups, explores their properties in the context of mapping class groups and Torelli groups, and investigates Galois actions and ramification in this setting.

Contribution

It introduces the concept of relative pro-ell completion, shows its non-injectivity for Torelli groups, and analyzes Galois representations associated with degenerations of algebraic curves.

Findings

Pro-ell completion of Torelli group does not inject into the relative pro-ell completion for genus ≥ 3.

The action of the pro-ell Torelli group on the fundamental group is not faithful for genus > 2.

Certain Galois representations are unramified at primes not equal to ell under specific conditions.

Abstract

Fix a prime number ell. In this paper we develop the theory of relative pro-ell completion of discrete and profinite groups -- a natural generalization of the classical notion of pro-ell completion -- and show that the pro-ell completion of the Torelli group does not inject into the relative pro-ell completion of the corresponding mapping class group when the genus is at least 3. As an application, we prove that when g > 2, the action of the pro-ell completion of the Torelli group T_{g,1} on the pro-ell fundamental group of a pointed genus g surface is not faithful. The choice of a first-order deformation of a maximally degenerate stable curve of genus g determines an action of the absolute Galois group G_Q on the relative pro-ell completion of the corresponding mapping class group. We prove that for all g all such representations are unramified at all primes \neq ell when the first order deformation is suitably chosen. This proof was communicated to us by Mochizuki and Tamagawa.