The triviality problem for profinite completions
Group Theory
2023-07-19 v3 Geometric Topology
Abstract
We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, this remains undecidable among the fundamental groups of compact, non-positively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperbolic groups, there cannot exist algorithms to determine largeness, the existence of a linear representation with infinite image (over any infinite field), or the rank of the profinite completion.
Cite
@article{arxiv.1401.2273,
title = {The triviality problem for profinite completions},
author = {Martin R. Bridson and Henry Wilton},
journal= {arXiv preprint arXiv:1401.2273},
year = {2023}
}
Comments
36 pages, 2 figures. Final version as accepted for publication, incorporating referee's comments. To appear in Inventiones Mathematicae