Banach fixed point property for Steinberg groups over commutative rings

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.