Farb fixed-point conjecture and Gromov Lp-cohomology conjecture

The problem links Farb’s fixed-point conjecture with Gromov’s $L^p$-cohomology conjecture for higher-rank arithmetic groups. Together they predict cohomological vanishing and strong restrictions on low-dimensional actions.

Known results

• López Neumann, 2023: proved substantial $L^p$-cohomology vanishing for several higher-rank groups, but not the full conjecture.
• $L^p$-cohomology and the geometry of $p$-harmonic forms$, 2024: proved the simple-group case for$1<p\le 2$ and degrees below real rank. - Viola, 2026: proved fixed points for specified higher-rank groups acting on certain infinite-dimensional non-Archimedean symmetric spaces. - Related work gives finite-image results for some low-dimensional manifold actions, not the acyclic-complex conclusion.

September 2026 claimed proof

A September 21, 2026 report attributes to Saar Bader, Shaked Bader, Uri Bader, and Roman Sauer a paper claiming vanishing below rank and finite-orbit results for actions on low-dimensional acyclic simplicial complexes, thereby resolving both conjectures. The claim awaits external peer review.

Current status (as of September 2026): Earlier partial results are established, while the combined resolution is only claimed in the new preprint and remains unverified.