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Groups Complexity Cryptology

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Hydra group doubles are not residually finite

Kristen Pueschel
  • Department of Mathematics, University of Arkansas, 309 SCEN, Fayetteville AR 72703, United States of America
  • :
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

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, October 2016

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