Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

CiteScore 2017: 0.32

SCImago Journal Rank (SJR) 2017: 0.208
Source Normalized Impact per Paper (SNIP) 2017: 0.322

Mathematical Citation Quotient (MCQ) 2017: 0.32

See all formats and pricing
More options …

Hydra group doubles are not residually finite

Kristen Pueschel
  • Corresponding author
  • Department of Mathematics, University of Arkansas, 309 SCEN, Fayetteville AR 72703, United States of America
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-10-13 | DOI: https://doi.org/10.1515/gcc-2016-0015


In 2013, Kharlampovich, Myasnikov, and Sapir constructed the first examples of finitely presented residually finite groups with large Dehn functions. Given any recursive function f, they produce a finitely presented residually finite group with Dehn function dominating f. There are no known elementary examples of finitely presented residually finite groups with super-exponential Dehn function. Dison and Riley’s hydra groups can be used to construct a sequence of groups for which the Dehn function of the kth group is equivalent to the kth Ackermann function. Kharlampovich, Myasnikov, and Sapir asked whether or not these groups are residually finite. We show that these constructions do not produce residually finite groups.

Keywords: Residual finiteness; hydra groups; Dehn function; separable subgroup

MSC 2010: 20E26; 20E06


  • [1]

    Baumslag B. and Tretkoff M., Residually finite HNN extensions, Comm. Algebra 6 (1978), no. 2, 179–194. Google Scholar

  • [2]

    Berlai F., Residual properties of free products, Comm. Algebra 44 (2016), 2959–2980. Google Scholar

  • [3]

    Bridson M. R. and Gersten S. M., The optimal isoperimetric inequality for torus bundles over the circle, Q. J. Math. 47 (1996), no. 185, 1–23. Google Scholar

  • [4]

    Burns R. G., Karrass A. and Solitar D., A note on groups with separable finitely generated subgroups, Bull. Aust. Math. Soc. 36 (1987), 153–160. Google Scholar

  • [5]

    Dison W., Einstein E. and Riley T. R., Taming the hydra: The word problem and extreme integer compression, preprint 2015, https://arxiv.org/abs/1509.02557.

  • [6]

    Dison W. and Riley T. R., Hydra groups, Comment. Math. Helv. 88 (2013), no. 3, 507–540. Google Scholar

  • [7]

    Kahrobaei D., Doubles of residually solvable groups, Aspects of Infinite Groups, Algebra Discrete Math. 1, World Scientific, Hackensack (2008), 192–200. Google Scholar

  • [8]

    Kharlampovich O., Myasnikov A. and Sapir M., Algorithmically complex residually finite groups, preprint 2013, https://arxiv.org/abs/1204.6506v5.

  • [9]

    Niblo G. A. and Wise D. T., The engulfing property for 3-manifolds, The Epstein Birthday Schrift Dedicated to David Epstein on the Occasion of his 60th Birthday, Geom. Topol. Monogr. 1, University of Warwick, Warwick (1998), 413–418. Google Scholar

  • [10]

    Niblo G. A. and Wise D. T., Subgroup separability, knot groups and graph manifolds, Proc. Amer. Math. Soc. 129 (2001), no. 3, 685–693. Google Scholar

About the article

Received: 2015-07-31

Published Online: 2016-10-13

Published in Print: 2016-11-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1444340

I gratefully acknowledge partial support from NSF grant DMS-1444340 and the hospitality of the Mathematical Institute, Oxford during the writing of this article.

Citation Information: Groups Complexity Cryptology, Volume 8, Issue 2, Pages 163–170, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2016-0015.

Export Citation

© 2016 by De Gruyter.Get Permission

Comments (0)

Please log in or register to comment.
Log in