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The triviality problem for profinite completions

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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 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.

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Acknowledgments

We first tried to prove Theorem A at the urging of Peter Cameron, who was interested in its implications for problems in combinatorics [12, 15]; we are grateful to him for this impetus. We thank Jack Button and Chuck Miller for stimulating conversations about Theorem A and its consequences. Finally, we are grateful for the insightful comments of the anonymous referee.

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Author notes

  1. Henry Wilton

    Present address: DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Oxford, OX2 6GG, UK

    Martin R. Bridson

  2. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

    Henry Wilton

Authors
  1. Martin R. Bridson
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  2. Henry Wilton
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Correspondence to Martin R. Bridson.

Additional information

This work was supported by Fellowships from the EPSRC (M. R. Bridson and H. Wilton) and by a Wolfson Research Merit Award from the Royal Society (M. R. Bridson).

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Bridson, M.R., Wilton, H. The triviality problem for profinite completions. Invent. math. 202, 839–874 (2015). https://doi.org/10.1007/s00222-015-0578-8

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  • DOI: https://doi.org/10.1007/s00222-015-0578-8

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