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 $$S$$, a compact orientable surface. Fix any characteristic subgroup $$H$$ of $$\pi_1(S)$$ and restrict to $$J(H)$$, any subgroup of mapping classes that induce the identity modulo $$H$$. To any unitary representation, $$r$$ of $$\pi_1(S)/H$$ we associate a higher-order $$\rho_r$$-invariant and a signature 2-cocycle $$\sigma_r$$. These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular each $$\rho_r$$ is a quasimorphism and each $$\sigma_r$$ is a bounded 2-cocycle on $$J(H)$$. In one of the simplest non-trivial cases, by varying $$r$$, we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the $$\rho_r$$ 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 $$S$$.