Projective simplicity and Bergman's property for unit groups of continuous rings
Friedrich Martin Schneider
TL;DR
The paper proves that for any non-discrete irreducible, continuous ring $R$, the projective unit group $\mathrm{PGL}(R)$ is simple and that $\mathrm{GL}(R)$ possesses uncountable strong cofinality and the Bergman property, leading to fixed-point properties for isometric actions on complete CAT(0) spaces and Serre properties $(FH)$ and $(FA)$. A key contribution is an abstract criterion for uncountable strong cofinality in unit groups of unital rings, built around corner rings and idempotent-driven subgroups, which is then instantiated to $\mathrm{GL}(R)$ via matrix-unit corners. By combining Bass’s results on $\mathrm{GL}_n$ with Bernard–Schneider’s commutator decompositions, the authors deduce bounded normal generation for $\mathrm{PGL}(R)$, answering Carderi and Thom’s questions. The results apply broadly to irreducible, non-discrete continuous rings, including operator-algebraic constructions, and yield a robust blend of algebraic and geometric consequences with potential applications to infinite-dimensional groups and operator rings. Overall, the work advances the understanding of simplicity and generation phenomena in unit groups of continuous rings and their projective counterparts.
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.