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Licensed Unlicensed Requires Authentication Published by De Gruyter April 15, 2014

On torsion in finitely presented groups

  • Maurice Chiodo EMAIL logo

Abstract.

We describe an algorithm that, on input of a recursive presentation P of a group, outputs a recursive presentation of a torsion-free quotient of P, isomorphic to P whenever P is itself torsion-free. Using this, we show the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed (first proved by Belegradek). We apply our techniques to show that recognising embeddability of finitely presented groups is Π20-hard, Σ20-hard, and lies in Σ30. We also show that the sets of orders of torsion elements of finitely presented groups are precisely the Σ20 sets which are closed under taking factors.

Funding source: University of Melbourne Overseas Research Experience Scholarship

Funding source: FIRB “Futuro in Ricerca”

Award Identifier / Grant number: RBFR10DGUA_002

Funding source: Swiss National Science Foundation

Award Identifier / Grant number: FN PP00P2-144681/1

The author wishes to thank Jack Button, Andrew Glass, Steffen Lempp, Vincenzo Marra, Chuck Miller and Rishi Vyas for their many useful conversations and comments which led to the overall improvement of this work. Thanks also go to François Dorais, Stefan Kohl, Benjamin Steinberg and Henry Wilton for their thoughtful discussion on MathOverflow, which led to the addition of Section 5 to this work. We thank Igor Belegradek for bringing the work [Internat. J. Algebra Comput. 18 (2008), no. 1, 97–110] to our attention, and to an anonymous referee for suggesting improvements. Finally, thanks must go to the late Greg Hjorth, whose suggestion of making contact with Steffen led to the eventual writing of this work.

Received: 2013-10-3
Published Online: 2014-4-15
Published in Print: 2014-5-1

© 2014 by Walter de Gruyter Berlin/Boston

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