On the complexity of epimorphism testing with virtually abelian targets

TL;DR

The paper investigates the complexity of testing for surjective homomorphisms from finitely presented groups to various virtually abelian target classes. By encoding epimorphism existence as systems of equations and translating these into polynomial-time matrix-span problems (MatrixSubspanA and MatrixSubspanB), the authors establish $\mathsf{NP}$-completeness for direct products of abelian and finite groups, virtually cyclic targets, and certain restricted semidirect products, and they extend $\mathsf{NP}$-hardness to fixed dihedral groups $D_{2n}$ with $n$ not a power of $2$. Core technical contributions include a reduction to matrix problems via $(Q,\tau)$-presentations and SpecialExt data, and the demonstration that the key matrix problems are solvable in polynomial time, placing the decision problems in $\mathsf{P}$ for the abelian-direct-product and related cases. The work broadly maps the landscape of epimorphism-testing complexity, showing sharp boundaries between decidability and hardness, and opening questions about uniform epimorphism over broader target classes and free-group targets. Overall, the paper advances the understanding of when epimorphism testing remains tractable and when it becomes intractable, with concrete constructions and reductions that tie group-theoretic questions to classical algorithmic linear-algebraic problems.

Abstract

Friedl and Löh (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.

Theorems & Definitions (119)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Lemma 1.1: von Dyck's lemma Benc2013
• Lemma 1.2
• Remark 1.3: Action is determined by $q$ when $N$ is abelian
• Remark 1.4
• Definition 1.5: $\mathrm{SpecialExt}$
• Definition 1.6: $\mathrm{RestrAbelSemi}$