Outer automorphism groups and the Atiyah Conjecture

Abstract

Let $G$ be the fundamental group of a compact surface, a finitely generated free group, or more generally a finitely generated right-angled Artin group. We prove that the von Neumann dimension function of $\mathrm{Out}(G)$ is valued in a discrete subgroup of $\mathbb Q$. This is accomplished by establishing the Strong Atiyah Conjecture for a torsion-free subgroup of $\mathrm{Out}(G)$ of finite index. We also prove that for every field $\mathbb K$, there exists a torsion-free subgroup $H \leqslant \mathrm{Out}(G)$ of finite index such that $\mathbb K[H]$ embeds into a division ring, and hence satisfies the Zero Divisor Conjecture. These results are obtained by establishing analogous ones for a suitable open subgroup of $\mathrm{Out}(\mathbf G)$ and its completed group algebra, where $\mathbf G$ denotes the pro-$p$ completion of $G$. In an appendix, the first author shows that an automorphism of a free nilpotent group is inner if and only if it induces an inner automorphism of its pro-$p$ completion.