Finding non-trivial elements and splittings in groups

TL;DR

This paper investigates the limitations of algorithmically identifying non-trivial elements, splittings, and embeddings in finitely presented groups, showing such procedures cannot exist in general, impacting group theory and topology.

Contribution

It proves the non-existence of general algorithms for producing non-trivial elements, decompositions, or embeddings in finitely presented groups, addressing open problems in group theory.

Findings

No general procedure to find a non-trivial element from a finite presentation.

No general method to decompose a finite presentation into non-trivial factors.

Splitting groups with more than one end cannot be made algorithmic.

Abstract

It is well known that the triviality problem for finitely presented groups is unsolvable; we ask the question of whether there exists a general procedure to produce a non-trivial element from a finite presentation of a non-trivial group. If not, then this would resolve an open problem by J. Wiegold: `Is every finitely generated perfect group the normal closure of one element?' We prove a weakened version of our question: there is no general procedure to pick a non-trivial generator from a finite presentation of a non-trivial group. We also show there is neither a general procedure to decompose a finite presentation of a non-trivial free product into two non-trivial finitely presented factors, nor one to construct an embedding from one finitely presented group into another in which it embeds. We apply our results to show that a construction by Stallings on splitting groups with more than one end can never be made algorithmic, nor can the process of splitting connect sums of non-simply connected closed 4-manifolds.