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Conjugacy search problem and the Andrews–Curtis conjecture

Conjugacy search problem and the Andrews–Curtis conjecture AbstractWe develop new computational methods for studying potential counterexamples to the Andrews–Curtis conjecture, in particular, Akbulut–Kurby examples AK⁡(n){\operatorname{AK}(n)}.We devise a number of algorithms in an attempt to disprove the most interesting counterexample AK⁡(3){\operatorname{AK}(3)}.That includes an efficient implementation of the folding procedure for pseudo-conjugacy graphs, based on the original modification of a classic disjoint-set data structure.To improve metric properties of the search space (the set of balanced presentations of the trivial group), we introduce a new transformation, called an ACM-move, that generalizes the original Andrews–Curtis transformations and discuss details of a practical implementation.To reduce growth of the search space, we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group.We prove that automorphism moves can be applied to Akbulut–Kurby presentations.The improved technique allows us to enumerate balanced presentations AC-equivalent to AK⁡(3){\operatorname{AK}(3)}with relations of lengths up to 20 (previous record was 17). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

Conjugacy search problem and the Andrews–Curtis conjecture

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Publisher
de Gruyter
Copyright
© 2019 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6104
eISSN
1869-6104
DOI
10.1515/gcc-2019-2005
Publisher site
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Abstract

AbstractWe develop new computational methods for studying potential counterexamples to the Andrews–Curtis conjecture, in particular, Akbulut–Kurby examples AK⁡(n){\operatorname{AK}(n)}.We devise a number of algorithms in an attempt to disprove the most interesting counterexample AK⁡(3){\operatorname{AK}(3)}.That includes an efficient implementation of the folding procedure for pseudo-conjugacy graphs, based on the original modification of a classic disjoint-set data structure.To improve metric properties of the search space (the set of balanced presentations of the trivial group), we introduce a new transformation, called an ACM-move, that generalizes the original Andrews–Curtis transformations and discuss details of a practical implementation.To reduce growth of the search space, we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group.We prove that automorphism moves can be applied to Akbulut–Kurby presentations.The improved technique allows us to enumerate balanced presentations AC-equivalent to AK⁡(3){\operatorname{AK}(3)}with relations of lengths up to 20 (previous record was 17).

Journal

Groups Complexity Cryptology – de Gruyter

Published: May 1, 2019

References

  • Dehn functions and l1l_{1}-norms of finite presentations
  • Balanced presentations of the trivial group
  • On the dunce hat
  • Balanced presentations of the trivial group
  • On Dehn’s algorithm and the conjugacy problem
  • Extended Nielsen transformations and the trivial group
  • On the algorithmic unsolvability of the word problem in group theory
  • A fast algorithm for Stallings’ folding process
  • Group presentations and formal deformations
  • The Zeeman conjecture for standard spines is equivalent to the Poincaré conjecture
  • On balanced presentations of the trivial group
  • Random van Kampen diagrams and algorithmic problems in groups
  • Extended Nielsen operations in free groups
  • Filling length in finitely presentable groups