The motivic Lie algebra embeds into the cohomology of the general linear group

Abstract

We show that the motivic Lie algebra of mixed Tate motives over $\mathbb{Z}$ embeds canonically into the unstable compactly-supported cohomology of locally symmetric spaces for $\mathrm{GL}_g(\mathbb{Z})$, and into the weight-zero compactly-supported cohomology of $\mathcal{A}_g$, the moduli space of principally polarized abelian varieties. Our construction passes through tropical geometry and graph complexes: compactly-supported analogues of the Borel classes pull back via the tropical Torelli map to canonical graph cocycles. The latter were recently identified with cocycles studied previously by Rossi and Willwacher. We combine their results with the theory of single-valued periods to conclude that the cocycles map to generators of the motivic Lie algebra. We also compute the canonical graph cocycles for all graphs with $\leq14$ edges.

25 pages, ancillary files with explicit graph cocycles