Vanishing of $\ell^2$-Betti numbers for inner amenable groups

Robin Tucker-Drob

arXiv:2608.08150v1 Primary: math.GR Secondary: math.OA

Abstract

We prove that every countable inner amenable group has vanishing $\ell^2$-Betti numbers in all degrees. This was previously open in every degree $n\geq2$.

Disclosure

“g seminar in Fall 2025, culminating in a proof of the Cheeger–Gromov theorem. This work was partially supported by NSF grant DMS-2246684. Disclosure of generative-AI tool use. In preparing this article, I used Claude Opus 5, Fable 5, and GPT-5.6 Sol Ultra for exploratory mathematical conversations and editorial feedback. Before using these tools, I had developed a degree-two argument using the prism operators under an additional technical hypothesis. Subsequent discussions with GP”
PDF page 2

Classification: Brainstorming or outlining
Multiplier: 2
Verified: 2026-08-23T05:01:00.949013Z

Structural counts

Pages 10 pdf

Theorems 1 source

Lemmas 1 source

Propositions 4 source

Corollaries 0 source

Definitions 0 source

Displayed equations 50 source

Bibliography entries 12 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file ia-l2-vanish.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.