2023/07/20 by Izhar Oppenheim, Oppenheim, Izhar
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2307.11064
openalex publication_date 2023/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main result of this paper is that all affine isometric actions of higher rank Steinberg groups over commutative rings on uniformly convex Banach spaces have a fixed point. We consider Steinberg groups over classical root systems and our analysis covers almost all such Steinberg groups excluding a single rank 2 case. The proof of our main result stems from two independent results - a result regarding relative fixed point properties of root subgroups of Steinberg groups and a result regarding passing from relative fixed point properties to a (global) fixed point property. The latter result is proven in the general setting of groups graded by root systems and provides a far reaching generalization of the work of Ershov, Jaikin-Zapirain and Kassabov who proved a similar result regarding property (T) for such groups. As an application of our main result, we give new constructions of super-expanders.