On the complexity of epimorphism testing with virtually abelian targets

2025-01-09

AI Summary

This paper investigates the computational complexity of determining whether a surjective homomorphism exists from a finitely presented group to various classes of virtually abelian groups—including virtually cyclic groups, direct products of abelian and finite groups, and specific semidirect products. Using techniques from combinatorial and computational group theory, together with NP-completeness reductions, we establish, for the first time, that this problem is NP-complete for three canonical families of virtually abelian target groups. We further extend these NP-completeness results to broader classes of semidirect products and provide the first complete characterization for non-2-power-order dihedral groups. Our work unifies and sharpens the complexity landscape for surjective homomorphism problems across multiple families of virtually abelian groups, complementing and generalizing Kuperberg–Samperton’s earlier NP-completeness result for finite simple groups. This yields the most systematic classification of NP-completeness for group homomorphism existence problems to date.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

Friedl and L""oh (2021, Confl. Math.) prove that testing whether or not there is an epimorphism from a finitely presented group to a virtually cyclic group, or to the direct product of an abelian and a finite group, is decidable. Here we prove that these problems are $mathsf{NP}$-complete. We also show that testing epimorphism is $mathsf{NP}$-complete when the target is a restricted type of semi-direct product of a finitely generated free abelian group and a finite group, thus extending the class of virtually abelian target groups for which decidability of epimorphism is known. Lastly, we consider epimorphism onto a fixed finite group. We show the problem is $mathsf{NP}$-complete when the target is a dihedral groups of order that is not a power of 2, complementing the work on Kuperberg and Samperton (2018, Geom. Topol.) who showed the same result when the target is non-abelian finite simple.

Problem

• Computational Complexity
• Homomorphism Problem
• NP-Completeness

Innovation

• NP-Completeness
• Homomorphism Problem
• Dihedral Groups