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 \(S\).