Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces

Abstract

We describe the group version of the trace maps given in [29]. This gives rise to abelian quotients of symplectic IA-automorphism groups of nilpotent quotients of the fundamental groups of compact surfaces. By making use of them, we construct a representation of the group $\mathcal{H}_{g,1}$ of homology cobordism classes of homology cylinders introduced by Garoufalidis and Levine [6]. We define various cohomology classes of $\mathcal{H}_{g,1}$ and propose a few problems concerning them. In particular, we mention a possible relation to additive invariants for the group $\Theta_{\mathbb{Z}}^{3}$ of homology cobordism classes of homology 3-spheres.