Geometric properties of unit groups of von Neumann's continuous rings
Friedrich Martin Schneider
TL;DR
This work studies the unit groups of von Neumann's continuous rings under the rank metric, building on the unique rank function and density of algebraic elements to develop a geometric/topological framework. It combines metric geometry (length spaces, geodesics), algebraic corner decompositions, and metric ultraproducts to show: (i) topological simplicity modulo the center for the projective unit group via bounded normal generation, (ii) a geodesic triangulation of algebraic elements that makes GL$(R)$ a length space with path- and local-path-connectedness, and (iii) escape-dynamics rigidity, i.e., absence of non-zero escape functions leading to strong automatic-continuity and representation-rigidity results. The results yield powerful constraints on actions and homomorphisms from GL$(R)$ into other groups, extend to ultraproducts, and provide uniform geodesic-width bounds, highlighting deep connections between algebraic structure and geometric/topological properties of unit groups in continuous rings.
Abstract
We prove that, if $R$ is a non-discrete irreducible, continuous ring, then its unit group $\mathrm{GL}(R)$, equipped with the topology generated by the rank metric, is topologically simple modulo its center, path-connected, locally path-connected, bounded in the sense of Bourbaki, and not admitting any non-zero escape function. All these topological insights are consequences of more refined geometric results concerning the rank metric, in particular with regard to the set of algebraic elements. Thanks to the phenomenon of automatic continuity, our results also have non-trivial ramifications for the underlying abstract groups.