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The complexity of solution sets to equations in hyperbolic groups

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Abstract

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, as shortlex geodesic words (or any regular set of quasigeodesic normal forms), is an EDT0L language whose specification can be computed in NSPACE(n2 log n) for the torsion-free case and NSPACE(n4 log n) in the torsion case. Furthermore, in the presence of effective quasi-isometrically embeddable rational constraints, we show that the full set of solutions to systems of equations in a hyperbolic group remains EDT0L.

Our work combines the geometric results of Rips, Sela, Dahmani and Guirardel on the decidability of the existential theory of hyperbolic groups with the work of computer scientists including Plandowski, Jeż, Diekert and others on PSPACE algorithms to solve equations in free monoids and groups using compression, and involves an intricate language-theoretic analysis.

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Acknowledgements

We sincerely thank Yago Antolin, Alex Bishop, François Dahmani, Volker Diekert, Derek Holt, Jim Howie and Alex Levine for invaluable help with this work. We also thank the anonymous reviewer for their careful reading and comments.

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Authors and Affiliations

  1. School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland

    Laura Ciobanu

  2. School of Mathematical and Physical Sciences, University of Technology Sydney, Ultimo, NSW, 2007, Australia

    Murray Elder

Authors
  1. Laura Ciobanu
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  2. Murray Elder
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Correspondence to Laura Ciobanu.

Additional information

Research supported by Australian Research Council (ARC) Project DP160100486, EPSRC grant EP/R035814/1, a Follow-On Grant from the International Centre of Mathematical Sciences (ICMS), Edinburgh, and an LMS Scheme 2 grant.

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Ciobanu, L., Elder, M. The complexity of solution sets to equations in hyperbolic groups. Isr. J. Math. 245, 869–920 (2021). https://doi.org/10.1007/s11856-021-2232-z

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  • DOI: https://doi.org/10.1007/s11856-021-2232-z

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