Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders

TL;DR

This paper introduces higher-order signature cocycles and rho-invariants for subgroups of the mapping class group, generalizing Meyer cocycles and revealing new structures and invariants in the study of surface homeomorphisms and homology cylinders.

Contribution

It defines new higher-order invariants for mapping class subgroups, generalizing Meyer cocycles, and demonstrates their properties and applications to homology cylinders.

Findings

Infinite families of linearly independent quasimorphisms and signature cocycles.

Certain invariants restrict to homomorphisms on specific subgroups.

Many invariants extend to the full mapping class group and homology cylinders.

Abstract

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.