The mapping class group invariants of the truncated group ring
Andreas Stavrou
TL;DR
The paper proves that the Γ_{g,1}-invariants of the Passi representations ${\mathcal P}_k=\mathbb{Q}\pi/\mathcal{I}^{k+1}$ are, in a stable range where $k+1\le 2g$, generated by two components: the boundary-loop part spanned by $((\zeta-1)^i+\mathcal{I}^{k+1})$ for $2i<k$, and the symplectic invariants $[H^{\otimes k}]^{\mathrm{Sp}_{2g}(\mathbb{Z})}$. The authors reduce the problem to invariants of the quotients $\mathcal{I}^k/\mathcal{I}^{k+2}$, which fit into an extension $H^{\otimes(k+1)} \to \mathcal{I}^k/\mathcal{I}^{k+2} \to H^{\otimes k}$, and analyze these using the Johnson homomorphism via the Torelli action. Central to the argument is a detailed study of the action of a Torelli element $\varphi$ on monomials in a symplectic basis, organized into chord-diagram invariants $\omega_C$, and showing that, up to $k$ in the stable range, the only invariant contribution beyond the boundary-part comes from $[H^{\otimes k}]^{\mathrm{Sp}_{2g}(\mathbb{Z})}$. This yields a cohomological computation for non-symplectic coefficients, extending prior work and informing higher-degree analyses via the same diagrammatic and Johnson-based framework. The methods combine invariant theory of chord diagrams, Fox's associated graded, and Johnson/Morita-type derivations to obtain precise structural descriptions of mapping class group invariants on truncated group rings, with potential implications for configuration spaces and related filtrations.
Abstract
We compute the invariant subspace of the rational group ring of a surface, truncated by powers of the augmentation ideal, under the action of the mapping class group. The surface is compact, oriented with one boundary component. This provides the first group cohomology computation for the mapping class group with non-symplectic coefficients since Kawazumi-Soulié. Our computation is valid in a range growing with the genus.