2025/10/08 by Nicholas G. Vlamis, Vlamis, Nicholas G. · 1 citation
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2510.07541
openalex publication_date 2025/10/08 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28
Word metrics on finitely generated groups have canonical quasi-isometry classes, making quasi-isometry invariants genuine group invariants. Rosendal generalized this phenomenon to topological groups through CB-generation, but in the general topological setting the resulting quasi-isometry invariants are not invariants of the underlying abstract group. Specializing to the discrete case yields what we call SB-generated groups, where the invariants are genuinely algebraic. We show that SB-generation arises naturally in transformation groups by identifying several broad families of examples: the identity component of homeomorphism groups of closed manifolds, certain big mapping class groups, and homeomorphism groups of compact well-ordered spaces with successor limit capacity. These results demonstrate that SB-generation provides a robust extension of finite generation.