Groupoid Extensions of Mapping Class Representations for Bordered Surfaces

TL;DR

This paper extends classical representations of the mapping class group of bordered surfaces to the broader context of the mapping class groupoid, providing explicit formulas and computational results for these extensions.

Contribution

It introduces a method to lift known mapping class group representations to the groupoid level, including explicit formulas and kernel-image analysis.

Findings

Extended symplectic representation rationally and integrally.

Computed kernel and image of the groupoid representation.

Applicable to various representations factoring through the fundamental group.

Abstract

The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upon choosing a basepoint. A combinatorial model for this, the mapping class groupoid, arises from the invariant cell decomposition of Teichm\"uller space, whose fundamental path groupoid is called the Ptolemy groupoid. It is natural to try to extend representations of the mapping class group to the mapping class groupoid, i.e., construct a homomorphism from the mapping class groupoid to the same target that extends the given representations arising from various choices of basepoint. Among others, we extend both aforementioned representations to the groupoid level in this sense, where the symplectic representation is lifted both rationally and integrally. The techniques of proof include several algorithms involving fatgraphs and chord diagrams. The former extension is given by explicit formulae depending upon six essential cases, and the kernel and image of the groupoid representation are computed. Furthermore, this provides groupoid extensions of any representation of the mapping class group that factors through its action on the fundamental group of the surface including, for instance, the Magnus representation and representations on the moduli spaces of flat connections.