The triviality problem for profinite completions

• 成果类型: Article
• 署名作者: Bridson, Martin R.; Wilton, Henry
• 署名单位: University of Oxford; University of London; University College London
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0578-8
• 发表日期: 2015
• 页码: 839-874
• 关键词: small cancellation groups; finite-groups; HYPERBOLIC GROUPS; geometry; complexes; polygons; infinite; graphs

摘要

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 property 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.