Projective simplicity and Bergman's property for unit groups of continuous rings

Abstract

We prove that the projective unit group $\mathrm{PGL}(R)$, i.e., the quotient of the unit group $\mathrm{GL}(R)$ modulo its center, of any non-discrete irreducible, continuous ring $R$ is simple. Moreover, we show that $\mathrm{GL}(R)$ has uncountable strong cofinality, that is, it is not the union of a countable chain of proper subgroups and it has finite width with respect to any generating set. Equivalently, every isometric action of $\mathrm{GL}(R)$ on a metric space has bounded orbits. It follows that every action of $\mathrm{GL}(R)$ by isometries on a non-empty complete $\mathrm{CAT}(0)$ space admits a fixed point. In particular, $\mathrm{GL}(R)$ possesses Serre's properties $(FH)$ and $(FA)$. Furthermore, our results entail that $\mathrm{PGL}(R)$ has bounded normal generation. In turn, we answer two questions by Carderi and Thom.