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The Isomorphism Problem for Plain Groups Is in Σ₃^{𝖯}

Authors Heiko Dietrich , Murray Elder , Adam Piggott , Youming Qiao , Armin Weiß

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Author Details

Heiko Dietrich
  • Monash University, Clayton, Australia
Murray Elder
  • University of Technology Sydney, Ultimo, Australia
Adam Piggott
  • Australian National University, Canberra, Australia
Youming Qiao
  • University of Technology Sydney, Ultimo, Australia
Armin Weiß
  • Universität Stuttgart, Germany

Funding

  • Dietrich, Heiko: Supported by Australian Research Council grant DP190100317.
  • Elder, Murray: Supported by Australian Research Council grant DP210100271.
  • Piggott, Adam: Supported by Australian Research Council grant DP210100271.
  • Qiao, Youming: Supported by Australian Research Council grant DP200100950.

Acknowledgements

We wish to thank the reviewers for their helpful comments and corrections.

Cite As Get BibTex

Heiko Dietrich, Murray Elder, Adam Piggott, Youming Qiao, and Armin Weiß. The Isomorphism Problem for Plain Groups Is in Σ₃^{𝖯}. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.STACS.2022.26

Abstract

Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. Sénizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in PSPACE when the input is given in terms of so-called virtually free presentations. Here we consider the isomorphism problem for the class of plain groups, that is, groups that are isomorphic to a free product of finitely many finite groups and finitely many copies of the infinite cyclic group. Every plain group is naturally and efficiently presented via an inverse-closed finite convergent length-reducing rewriting system. We prove that the isomorphism problem for plain groups given in this form lies in the polynomial time hierarchy, more precisely, in Σ₃^𝖯. This result is achieved by combining new geometric and algebraic characterisations of groups presented by inverse-closed finite convergent length-reducing rewriting systems developed in recent work of the second and third authors (2021) with classical finite group isomorphism results of Babai and Szemerédi (1984).

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Rewrite systems
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Computability
Keywords
  • plain group
  • isomorphism problem
  • polynomial hierarchy
  • Σ₃^{𝖯} complexity class
  • inverse-closed finite convergent length-reducing rewriting system

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