Groupoid Lifts of Mapping Class Representations for Bordered Surfaces

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 arises from the invariant cell decomposition of Teichmüller space, whose fundamental path groupoid is called the Ptolemy groupoid. It is natural to try to lift representations of the mapping class group to the Ptolemy groupoid, i.e., construct a homomorphism from the Ptolemy groupoid to the same target so that if a path in Teichmüller space covers a loop in moduli space, then the two representations coincide. We lift both aforementioned representations to the groupoid level in this sense. The techniques of proof include fatgraphs, chord diagrams, and their relationship. The former lift is given by explicit formulae depending upon six essential cases, and the kernel and image of the groupoid representaion is computed. Furthermore, this provides groupoid lifts of representations of the mapping class group that factor 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.