Width bounds and Steinhaus property for unit groups of continuous rings

TL;DR

This paper establishes algebraic and topological properties of the unit group of continuous rings, showing it can be decomposed into products of commutators and involutions, and possesses the Steinhaus property, ensuring the continuity of homomorphisms.

Contribution

It provides a new algebraic decomposition theorem for the unit group of continuous rings and proves the Steinhaus property, leading to implications for homomorphism continuity and dynamical fixed point results.

Findings

Every element of the unit group is a product of 7 commutators.

Every element of the unit group is a product of 16 involutions.

The unit group has the Steinhaus property with respect to the rank topology.

Abstract

We prove an algebraic decomposition theorem for the unit group $\mathrm{GL}(R)$ of an arbitrary non-discrete irreducible, continuous ring $R$ (in von Neumann's sense), which entails that every element of $\mathrm{GL}(R)$ is both a product of $7$ commutators and a product of $16$ involutions. Combining this with further insights into the geometry of involutions, we deduce that $\mathrm{GL}(R)$ has the so-called Steinhaus property with respect to the natural rank topology, thus every homomorphism from $\mathrm{GL}(R)$ to a separable topological group is necessarily continuous. Due to earlier work, this has further dynamical ramifications: for instance, for every action of $\mathrm{GL}(R)$ by homeomorphisms on a non-void metrizable compact space, every element of $\mathrm{GL}(R)$ admits a fixed point in the latter. In particular, our results answer two questions by Carderi and Thom, even in generalized form.