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. Fix any characteristic subgroup H C π1(Σ) and restrict to J(H), any subgroup of mapping classes that induce the identity on π1(Σ)/H. To any unitary representation ψ of π1(Σ)/H is associated 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. We make calculations in the simplest non-trivial case and show that, by varying ψ, we get infinite families of linearly independent quasimorphisms and signature cocycles. We introduce new subgroups of the Torelli group and show that the ρψ restrict to homomorphisms on these subgroups. Many of these invariants extend to the full mapping class group and some extend to the monoid of homology cylinders based on Σ.