Completions of mapping class groups and the cycle $C - C^-$

TL;DR

This paper investigates the proalgebraic completions of mapping class groups, revealing a non-trivial central extension related to the Jacobian of algebraic curves, with implications for understanding their algebraic and geometric structures.

Contribution

It establishes that the unipotent completion of the Torelli group forms a non-trivial central extension of the prounipotent radical of the mapping class group's symplectic completion for genus at least 8.

Findings

The unipotent completion of the Torelli group extends centrally by Q.

The extension is linked to the line bundle from the archemidean height of a cycle.

Basic theory of relative completions is developed.

Abstract

In this paper we study the proalgebraic completion of mapping class relative to their maps to the symplectic group. The main result is that the natural map from the unipotent (a.k.a. Malcev) completion of the Torelli group to the prounipotent radical of the Sp_g completion of the mapping class group is a non trivial central extension with kernel isomorphic to Q, at least when g \ge 8. The theorem is proved by relating the central extension to the line bundle associated to the archemidean height of the cycle C - C- in the Jacobian of the curve C. We also develop some of the basic theory of relative completions.