Helly-type theorems, CAT(0) spaces, and actions of Aut(F_n)

Journal article

We prove a variety of fixed-point theorems for groups acting on CAT$(0)$ spaces. Fixed points are obtained by a bootstrapping technique, whereby increasingly large subgroups are proved to have fixed points: specific configurations in the subgroup lattice of $\Gamma$ are exhibited and Helly-type theorems are developed to prove that the fixed-point sets of the subgroups in the configuration intersect. In this way, we obtain lower bounds on the smallest dimension $\mathrm{FixDim}(\Gamma)+1$ in which various groups of geometric interest can act on a complete CAT$(0)$ space without a global fixed point. For automorphism groups of free groups, we prove $\mathrm{FixDim}(\mathrm{Aut}(F_n)) \geq \lfloor \tfrac{2n}{3} \rfloor$.

Authors: Bridson, MR

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Advances in Mathematics
• Volume: 475
• Article number: 110343
• Publication date: 2025-05-16
• Acceptance date: 2025-05-01
• DOI: 10.1016/j.aim.2025.110343
• EISSN: 1090-2082
• ISSN: 0001-8708
• Language: English
• Keywords: Helly-type theorems; fixed-point theorems; CAT(0) spaces; automorphism groups of free groups