Finding non-trivial elements and splittings in groups
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.
Keywords
Cite
@article{arxiv.1002.2786,
title = {Finding non-trivial elements and splittings in groups},
author = {Maurice Chiodo},
journal= {arXiv preprint arXiv:1002.2786},
year = {2012}
}
Comments
16 pages, revised version, minor corrections made, additional results on algorithms to construct embeddings and enumerate subgroups. v3: minor changes to typesetting and theorem numberings, minor corrections made