Higher-Order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders

We define families of invariants for elements of the mapping class group of Σ, a compact-orientable surface. For any characteristic subgroup $\triangleleft \pi_1(\Sigma)$, let J(H) denote the subgroup of mapping classes that induce the identity on π1(Σ)/H. To any unitary representation ψ of π1(Σ)/H, we associate a higher-order ρψ-invariant and a signature 2-cocycle σψ. These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular, each ρψ is a quasimorphism and each σψ is a bounded 2-cocycle on J(H). In one of the simplest nontrivial cases, by varying ψ, we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the ρψ restrict to homomorphisms on certain interesting subgroups. Many of these invariants extend naturally to the full mapping class group and some extend to the monoid of homology cylinders based on Σ.