Banach fixed point property for Steinberg groups over commutative rings

TL;DR

This paper proves that higher rank Steinberg groups over commutative rings have fixed points when acting on uniformly convex Banach spaces, extending fixed point properties and enabling new super-expander constructions.

Contribution

It establishes fixed point properties for Steinberg groups over rings on Banach spaces, generalizing previous results and introducing a new approach for groups graded by root systems.

Findings

All affine isometric actions of certain Steinberg groups have fixed points.

A new method relates relative fixed point properties to global fixed point properties.

Application to constructing new super-expanders.

Abstract

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.