Epimorphism classes and relatively maximal metrics in large-scale geometry

2025/12/15 by Robert L. Tang, Tang, Robert
Mathematics · #20F65 #51F30 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Category Theory (math.CT) #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2512.13378

openalex publication_date 2025/12/15 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28

Abstract

We consider epimorphisms and several variant notions -- split, effective, regular, strong, and extremal -- and determine which of these coincide in the metric coarse and coarsely Lipschitz categories. In particular, we characterise extremal epis in the coarsely Lipschitz category via a relative maximality condition on the codomain metric; this can be viewed as a morphism-relative analogue of Rosendal's maximal metrics for topological groups.

Citations

• The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spaces
• Categorical characterisations of quasi-isometric embeddings
• An Invitation to Coarse Groups
• Large-scale geometry of the saddle connection graph
• Coarse quotients of metric spaces and embeddings of uniform Roe algebras
• Coarse quotients by group actions and the maximal Roe algebra
• Large scale geometry of homeomorphism groups
• Coarse Quotient Mappings between Metric Spaces
• Groups quasi-isometric to symmetric spaces

Related

• Maximal metric surfaces and the Sobolev-to-Lipschitz property
• Note on the induced metric on power sets of groups
• A maximal element of a moduli space of Riemannian metrics
• Large scale geometry of metrisable groups
• The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spaces