Outer automorphism groups and the Atiyah Conjecture

Mathematics > Group Theory [Submitted on 17 Jun 2026] Title:Outer automorphism groups and the Atiyah Conjecture View PDF HTML (experimental)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. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.