The epimorphism relation among countable groups is a complete analytic quasi-order
Su Gao, Feng Li, André Nies, Gianluca Paolini
TL;DR
The paper addresses the problem of classifying surjective homomorphisms between countable groups using Borel complexity theory. It first establishes analytic completeness for the epimorphism relation on countable pointed reflexive graphs via a result of Louveau and Rosendal and a Borel reduction, then transfers this complexity to countable groups through a novel construction based on countably generated Coxeter groups. The main result shows that the epimorphism relation on countable groups is a complete analytic quasi-order, implying that bi-epimorphism is a complete analytic equivalence relation. This work connects graph-theoretic reduction techniques with group-theoretic constructions, advancing the understanding of descriptive-set-theoretic complexity in algebraic structures and leaving open the abelian case.
Abstract
We prove that the epimorphism relation is a complete analytic quasi-order on the space of countable groups. In the process we obtain the result of indepent interest showing that the epimorphism relation on pointed reflexive graph is complete.