Geometric properties of unit groups of von Neumann's continuous rings

2025/09/01 by Friedrich Martin Schneider, Schneider, Friedrich Martin · 1 citation
Mathematics · #06C20 #16E50 #20E45 #20F70 #22A10 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2509.01556

openalex publication_date 2025/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that, if R is a non-discrete irreducible, continuous ring, then its unit group 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.

Citations

• Width bounds and Steinhaus property for unit groups of continuous rings
• Bounded normal generation is not equivalent to topological bounded normal generation
• Some geometric properties of metric ultraproducts of finite simple groups
• An exotic group as limit of finite special linear groups
• Lamplighter groups and von Neumann's continuous regular rings
• Embedding group algebras into finite von Neumann regular rings
• Turbulence, amalgamation and generic automorphisms of homogeneous structures
• Sofic groups and direct finiteness

Cited by

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

Related

• Width bounds and Steinhaus property for unit groups of continuous rings
• Projective simplicity and Bergman's property for unit groups of continuous rings
• Some notes on topological rings and their groups of units
• Lamplighter groups and von Neumann's continuous regular rings
• A generalization of Dirichlet's unit theorem