On the complexity of epimorphism testing with virtually abelian targets

TL;DR

This paper proves that testing for epimorphisms from finitely presented groups to various virtually abelian and related groups is NP-complete, extending known decidability results to computational complexity classifications.

Contribution

It establishes NP-completeness for epimorphism testing to a broader class of virtually abelian groups, including certain semi-direct products and fixed finite groups.

Findings

Proves NP-completeness for virtually cyclic targets.

Extends NP-completeness to semi-direct product targets.

Shows NP-completeness for epimorphisms onto certain dihedral groups.

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.