Hydra group doubles are not residually finite
Abstract
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.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1444340
Funding statement: I gratefully acknowledge partial support from NSF grant DMS-1444340 and the hospitality of the Mathematical Institute, Oxford during the writing of this article.
Acknowledgements
I want to thank my advisor Tim Riley for his help, suggestions, and corrections, Jason Manning for feedback on this work, and Mark Sapir, for a helpful conversation.
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© 2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Computing discrete logarithms using 𝒪((log q)2) operations from {+,-,×,÷,&}
- A parallel evolutionary approach to solving systems of equations in polycyclic groups
- Authenticated commutator key agreement protocol
- On the covering number of small symmetric groups and some sporadic simple groups
- Compositions of linear functions and applications to hashing
- Hydra group doubles are not residually finite
- The status of polycyclic group-based cryptography: A survey and open problems
- On irreducible algebraic sets over linearly ordered semilattices
- A nonlinear decomposition attack
Artikel in diesem Heft
- Frontmatter
- Computing discrete logarithms using 𝒪((log q)2) operations from {+,-,×,÷,&}
- A parallel evolutionary approach to solving systems of equations in polycyclic groups
- Authenticated commutator key agreement protocol
- On the covering number of small symmetric groups and some sporadic simple groups
- Compositions of linear functions and applications to hashing
- Hydra group doubles are not residually finite
- The status of polycyclic group-based cryptography: A survey and open problems
- On irreducible algebraic sets over linearly ordered semilattices
- A nonlinear decomposition attack
