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Marked fatgraph complexes and surface automorphisms

Combinatorial aspects of the Torelli-Johnson-Morita theory of surface automorphisms are extended to certain subgroups of the mapping class groups. These subgroups are defined relative to a specified homomorphism from the fundamental group of the surface onto an arbitrary group $K$. For $K$ abelian, there is a combinatorial theory akin to the classical case, for example, providing an explicit cocycle representing the first Johnson homomophism with target $Λ^3 K$. Furthermore, the Earle class with coefficients in $K$ is represented by an explicit cocyle.