Secondary characteristic classes for subgroups of automorphism groups of free groups

TL;DR

This paper introduces new secondary characteristic classes for subgroups of automorphism groups of free groups, linking them to known torsion invariants and proposing conjectural geometric interpretations.

Contribution

It defines three series of secondary classes for automorphism subgroups, connecting them to higher torsions and Morita classes, with new group cocycles and conjectural geometric meanings.

Findings

Secondary classes for IA-automorphism groups match Igusa's higher FR torsions.

Classes for mapping class groups are multiples of Mumford-Morita-Miller classes.

Proposes conjectural geometric interpretation of Morita classes.

Abstract

By analyzing how the Borel regulator classes vanish on various groups related to $\mathrm{GL}(n,\mathrm{Z})$, we define three series of secondary characteristic classes for subgroups of automorphism groups of free groups. The first case is the $\mathrm{IA}$-automorphism groups and we show that our classes coincide with higher $\mathrm{FR}$ torsions due to Igusa. The second case is the mapping class groups and our classes also turn out to be his higher torsions which are non-zero multiples of the Mumford-Morita-Miller classes of even indices. Our construction gives new group cocycles for these still mysterious classes. The third case is the outer automorphism groups of free groups of specific ranks. Here we give a conjectural geometric meaning to a series of unstable homology classes called the Morita classes. We expect that certain unstable secondary classes would detect them.