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arxiv: 2605.30535 · v1 · pith:DFZPP2LBnew · submitted 2026-05-28 · 🧮 math.GR

Undecidable Diophantine problems in generalisations of one-relator groups

Robert D. Gray , Alex Levine This is my paper

Pith reviewed 2026-06-28 23:43 UTC · model grok-4.3

classification 🧮 math.GR
keywords Diophantine problemone-relator groupsundecidabilityBaumslag-Solitar groupsfree-by-free groupsquasi-isometryone-relator productsgroup equations
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The pith

There exist generalised Baumslag-Solitar groups with an undecidable Diophantine problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines decidability of the Diophantine problem across families of groups that generalize one-relator groups in structure or geometry. It constructs explicit examples showing that the problem is undecidable in a generalised Baumslag-Solitar group, in a one-relator product of cyclic groups, and in a free-by-free group of the form F3 ⋊ F2. It further derives an undecidable example that is quasi-isometric to a one-relator group and shows undecidability of the constrained Diophantine problem for some one-relator group relative to a fixed free subgroup. A sympathetic reader would care because these results place concrete limits on where decidability can be expected to hold uniformly.

Core claim

The authors prove that there is a generalised Baumslag-Solitar group with an undecidable Diophantine problem. Using this example they obtain a group with an undecidable Diophantine problem that is quasi-isometric to a one-relator group. They also prove that there is a one-relator product of cyclic groups with an undecidable Diophantine problem and that there is a free-by-free group of the form F3 ⋊ F2 with an undecidable Diophantine problem. In addition they show that there is a one-relator group G containing a single fixed finite-rank free subgroup H such that the Diophantine problem for G with H-constraints is undecidable.

What carries the argument

Explicit constructions of groups in the classes of generalised Baumslag-Solitar groups, one-relator products of cyclic groups, and semidirect products F3 ⋊ F2 that reduce instances of known undecidable Diophantine problems to the equation problem in the constructed group.

If this is right

  • There is a generalised Baumslag-Solitar group with an undecidable Diophantine problem.
  • There is a group with an undecidable Diophantine problem that is quasi-isometric to a one-relator group.
  • There is a one-relator product of cyclic groups with an undecidable Diophantine problem.
  • There is a free-by-free group of the form F3 ⋊ F2 with an undecidable Diophantine problem.
  • There is a one-relator group G with a fixed finite-rank free subgroup H such that the Diophantine problem for G with H-constraints is undecidable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Decidability of the Diophantine problem, if it holds for all one-relator groups, must rely on features not shared by these nearby classes.
  • Quasi-isometry fails to preserve decidability of the Diophantine problem between the constructed group and its one-relator counterpart.
  • The same style of construction could be tested on the remaining open case of free-by-cyclic groups.
  • The constrained version indicates that the presence of a fixed free subgroup can introduce undecidability even inside an otherwise one-relator group.

Load-bearing premise

The specific group constructions admit reductions from known undecidable Diophantine instances while preserving the group-theoretic properties claimed for each family.

What would settle it

An algorithm that solves all equations in the particular generalised Baumslag-Solitar group constructed in the paper, or a demonstration that the reduction from an undecidable source problem fails to embed into that group.

read the original abstract

Motivated by the open problem of whether all one-relator groups have decidable Diophantine problem, in this paper we prove a collection of undecidability results about the Diophantine problem for several families of groups that are close to one-relator groups in various ways. We prove that there is a generalised Baumslag--Solitar group with an undecidable Diophantine problem. Using our example we show there is a group with an undecidable Diophantine problem that is quasi-isometric to a one-relator group. Also, we prove that there is a one-relator product of cyclic groups with an undecidable Diophantine problem. In addition, we show that there there is a one-relator group $G$, with a single fixed finite rank free subgroup $H$, such that the Diophantine problem for $G$ with $H$-constraints is undecidable. The related open question of whether there is a free-by-cyclic group with undecidable Diophantine problem is also discussed, and we prove that there is a free-by-free group of the form $F_3 \rtimes F_2$ with an undecidable Diophantine problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves undecidability of the Diophantine problem (DP) for several families of groups generalizing one-relator groups. It constructs a generalized Baumslag-Solitar group with undecidable DP, uses this to obtain a group with undecidable DP that is quasi-isometric to a one-relator group, exhibits a one-relator product of cyclic groups with undecidable DP, shows undecidability of the DP with constraints from a fixed finite-rank free subgroup H in a one-relator group, and constructs an F3 ⋊ F2 group with undecidable DP. All results are obtained via explicit reductions from undecidable instances of Hilbert's tenth problem over rings of integers in number fields, with the target groups shown to lie in the claimed classes.

Significance. If the reductions and group-theoretic verifications hold, the results are significant: they establish that undecidability persists under several natural generalizations of one-relator groups (including quasi-isometry and constrained variants), while leaving the free-by-cyclic case open. The explicit presentations and computable reductions from number-theoretic undecidability constitute a concrete contribution to the boundary between decidable and undecidable Diophantine problems in geometric group theory.

minor comments (2)
  1. Abstract, line beginning 'Also, we prove...': the phrase 'there there is' is a typographical error and should be corrected to 'there is'.
  2. The manuscript would benefit from a brief explicit statement (perhaps in the introduction) of the precise definition of the Diophantine problem used throughout, including the precise notion of 'solution set' preserved by the reductions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its undecidability claims via explicit group presentations and computable reductions from externally known undecidable Diophantine instances (Hilbert's tenth problem over rings of integers in number fields) to the Diophantine problem in generalised Baumslag-Solitar groups, one-relator products of cyclics, H-constrained one-relator groups, and F3 ⋊ F2. These reductions are shown to preserve solution sets while keeping the constructed groups inside the claimed families, with no parameter fitting, self-definitional equations, or load-bearing self-citations that collapse the argument to the paper's own inputs. The derivations remain independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard facts from combinatorial group theory about one-relator groups, Baumslag-Solitar groups, and known undecidable Diophantine instances in other groups; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • standard math Standard properties of one-relator groups, generalised Baumslag-Solitar groups, and free-by-free groups hold as stated in the combinatorial group theory literature.
    Invoked to ensure the constructed groups belong to the claimed classes.
  • domain assumption There exist groups with undecidable Diophantine problem from which reductions can be performed.
    Required for the undecidability transfer arguments.

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discussion (0)

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Reference graph

Works this paper leans on

98 extracted references · 11 canonical work pages · 1 internal anchor

  1. [1]

    https://mathoverflow.net/questions/160161/infinitely-many-finitely-generated-groups-having-the-same- cayley-graph/160178#160178

  2. [2]

    Alp¨ oge, M

    L. Alp¨ oge, M. Bhargava, W. Ho, and A. Shnidman. Rank stability in quadratic extensions and Hilbert’s tenth problem for the ring of integers of a number field.Invent. Math., 243(3):1129–1139, 2026

    2026

  3. [3]

    Antol´ ın and I

    Y. Antol´ ın and I. Foniqi. Intersection of parabolic subgroups in even Artin groups of FC-type.Proc. Edinb. Math. Soc. (2), 65(4):938–957, 2022

    2022

  4. [4]

    Ascari, M

    D. Ascari, M. Casals-Ruiz, and I. Kazachkov. On the isomorphism problem for generalized Baumslag-Solitar groups: invariants and flexible configurations, 2025. arXiv:2508.02306

    work page arXiv 2025

  5. [5]

    D. A. Barkauskas.Centralizers in Fundamental Groups of Graphs of Groups. PhD thesis, 2003

    2003

  6. [6]

    Baumslag, J

    G. Baumslag, J. W. Morgan, and P. B. Shalen. Generalized triangle groups.Math. Proc. Cambridge Philos. Soc., 102(1):25–31, 1987

    1987

  7. [7]

    Baumslag, A

    G. Baumslag, A. G. Myasnikov, and V. Shpilrain. Open problems in combinatorial group theory.Contem- porary Mathematics, 296:1–38, 2002

    2002

  8. [8]

    Bogopolski, A

    O. Bogopolski, A. Martino, O. Maslakova, and E. Ventura. The conjugacy problem is solvable in free-by- cyclic groups.Bull. London Math. Soc., 38(5):787–794, 2006

    2006

  9. [9]

    Bogopolski, A

    O. Bogopolski, A. Martino, and E. Ventura. Orbit decidability and the conjugacy problem for some exten- sions of groups.Trans. Amer. Math. Soc., 362(4):2003–2036, 2010

    2003

  10. [10]

    Bogopolski.Introduction to group theory

    O. Bogopolski.Introduction to group theory. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Z¨ urich, 2008. Translated, revised and expanded from the 2002 Russian original

    2008

  11. [11]

    V. V. Borisov. Simple examples of groups with unsolvable word problem.Mat. Zametki, 6:521–532, 1969

    1969

  12. [12]

    M. R. Bridson, T. R. Riley, and A. W. Sale. Conjugacy in a family of free-by-cyclic groups.arXiv preprint arXiv:2506.01248, 2025

    work page arXiv 2025

  13. [13]

    S. D. Brodski˘ ı. Equations over groups, and groups with one defining relation.Sibirsk. Mat. Zh., 25(2):84–103, 1984

    1984

  14. [14]

    J. R. B¨ uchi and S. Senger. Definability in the existential theory of concatenation and undecidable extensions of this theory.Z. Math. Logik Grundlag. Math., 34(4):337–342, 1988

    1988

  15. [15]

    Burillo and M

    J. Burillo and M. Elder. Metric properties of Baumslag-Solitar groups.Internat. J. Algebra Comput., 25(5):799–811, 2015

    2015

  16. [16]

    R. G. Burns, A. Karrass, and D. Solitar. A note on groups with separable finitely generated subgroups.Bull. Austral. Math. Soc., 36(1):153–160, 1987

    1987

  17. [17]

    Cadilhac, D

    M. Cadilhac, D. Chistikov, and G. Zetzsche. Rational Subsets of Baumslag-Solitar Groups. In A. Czumaj, A. Dawar, and E. Merelli, editors,47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), volume 168 ofLeibniz International Proceedings in Informatics (LIPIcs), pages 116:1–116:16, Dagstuhl, Germany, 2020. Schloss Dagstuhl – ...

    2020

  18. [18]

    C. M. Campbell, P. M. Heggie, E. F. Robertson, and R. M. Thomas. Cyclically presented groups embedded in one-relator products of cyclic groups.Proc. Amer. Math. Soc., 118(2):401–408, 1993

    1993

  19. [19]

    Casals-Ruiz and I

    M. Casals-Ruiz and I. Kazachkov. On systems of equations over free partially commutative groups.Mem. Amer. Math. Soc., 212(999):viii+153, 2011

    2011

  20. [20]

    Casals-Ruiz, I

    M. Casals-Ruiz, I. Kazachkov, and J. de la Nuez Gonz´ alez. On the elementary theory of graph products of groups.Dissertationes Math., 595:92, 2024

    2024

  21. [21]

    Casals-Ruiz, I

    M. Casals-Ruiz, I. Kazachkov, and A. Zakharov. Commensurability of Baumslag-Solitar groups.Indiana Univ. Math. J., 70(6):2527–2555, 2021

    2021

  22. [22]

    Ciobanu, A

    L. Ciobanu, A. Evetts, and A. Levine. Effective equation solving, constraints, and growth in virtually abelian groups.SIAM J. Appl. Algebra Geom., 9(1):235–260, 2025

    2025

  23. [23]

    Ciobanu and A

    L. Ciobanu and A. Garreta. Group equations with abelian predicates.Int. Math. Res. Not. IMRN, (5):4119– 4159, 2024. UNDECIDABLE DIOPHANTINE PROBLEMS 29

    2024

  24. [24]

    Ciobanu, R

    L. Ciobanu, R. D. Gray, and A. Levine. The Diophantine problem with abelianisation constraints in some free-by-cyclic groups.In preparation

  25. [25]

    Ciobanu, D

    L. Ciobanu, D. Holt, and S. Rees. Equations in groups that are virtually direct products.J. Algebra, 545:88– 99, 2020

    2020

  26. [26]

    M. M. Cohen and M. Lustig. On the dynamics and the fixed subgroup of a free group automorphism.Invent. Math., 96(3):613–638, 1989

    1989

  27. [27]

    D. J. Collins. Generation and presentation of one-relator groups with centre.Mathematische Zeitschrift, 157(1):63–77, 1977

    1977

  28. [28]

    F. Dahmani. Existential questions in (relatively) hyperbolic groups.Israel J. Math., 173:91–124, 2009

    2009

  29. [29]

    Dahmani and V

    F. Dahmani and V. Guirardel. Foliations for solving equations in groups: free, virtually free, and hyperbolic groups.J. Topol., 3(2):343–404, 2010

    2010

  30. [30]

    F. Dahmani. Existential questions in (relatively) hyperbolic groups.Israel Journal of Mathematics, 173:91– 124, 2009

    2009

  31. [31]

    Dahmani and R

    F. Dahmani and R. Li. Relative hyperbolicity for automorphisms of free products and free groups.J. Topol. Anal., 14(1):55–92, 2022

    2022

  32. [32]

    M. B. Day and R. D. Wade. Relative automorphism groups of right-angled Artin groups.J. Topol., 12(3):759– 798, 2019

    2019

  33. [33]

    de la Harpe.Topics in geometric group theory

    P. de la Harpe.Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000

    2000

  34. [34]

    Diekert and A

    V. Diekert and A. Muscholl. Solvability of equations in free partially commutative groups is decidable. InAutomata, languages and programming, volume 2076 ofLecture Notes in Comput. Sci., pages 543–554. Springer, Berlin, 2001

    2076

  35. [35]

    Dison and T

    W. Dison and T. R. Riley. Hydra groups.Comment. Math. Helv., 88(3):507–540, 2013

    2013

  36. [36]

    R. Dong. Linear equations with monomial constraints and decision problems in abelian-by-cyclic groups. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1892–1908. SIAM, Philadelphia, PA, 2025

    2025

  37. [37]

    Duchin, H

    M. Duchin, H. Liang, and M. Shapiro. Equations in nilpotent groups.Proc. Amer. Math. Soc., 143(11):4723– 4731, 2015

    2015

  38. [38]

    Duncan, A

    A. Duncan, A. Evetts, D. F. Holt, and S. Rees. Using EDT0L systems to solve some equations in the solvable Baumslag-Solitar groups.J. Algebra, 630:434–456, 2023

    2023

  39. [39]

    A. J. Duncan and J. Howie. The genus problem for one-relator products of locally indicable groups.Math. Z., 208(2):225–237, 1991

    1991

  40. [40]

    A. J. Duncan and J. Howie. Weinbaum’s conjecture on unique subwords of nonperiodic words.Proc. Amer. Math. Soc., 115(4):947–954, 1992

    1992

  41. [41]

    A. J. Duncan and J. Howie. One relator products with high-powered relators. InGeometric group theory, Vol. 1 (Sussex, 1991), volume 181 ofLondon Math. Soc. Lecture Note Ser., pages 48–74. Cambridge Univ. Press, Cambridge, 1993

    1991

  42. [42]

    A. J. Duncan and A. Juh´ asz. One-relator quotients of right-angled Artin groups.J. Algebra, 622:506–555, 2023

    2023

  43. [43]

    Elliott and A

    L. Elliott and A. Levine. The Diophantine problem in Thompson’s groupF.Math. Comp., 95(360):2013– 2023, 2026

    2013

  44. [44]

    J. L. Erˇ sov. Elementary group theories.Dokl. Akad. Nauk SSSR, 203:1240–1243, 1972

    1972

  45. [45]

    B. Fine, J. Howie, and G. Rosenberger. One-relator quotients and free products of cyclics.Proc. Amer. Math. Soc., 102(2):249–254, 1988

    1988

  46. [46]

    Fine and G

    B. Fine and G. Rosenberger.Algebraic generalizations of discrete groups, volume 223 ofMonographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 1999. A path to combinatorial group theory through one-relator products

    1999

  47. [47]

    Friedl and H

    S. Friedl and H. Wilton. The membership problem for 3-manifold groups is solvable.Algebr. Geom. Topol., 16(4):1827–1850, 2016

    2016

  48. [48]

    Garreta and R

    A. Garreta and R. D. Gray. On equations and first-order theory of one-relator monoids.Inform. and Comput., 281:Paper No. 104745, 19, 2021

    2021

  49. [49]

    Garreta, A

    A. Garreta, A. Miasnikov, and D. Ovchinnikov. Random nilpotent groups, polycyclic presentations, and Diophantine problems.Groups Complex. Cryptol., 9(2):99–115, 2017

    2017

  50. [50]

    Garreta, A

    A. Garreta, A. Miasnikov, and D. Ovchinnikov. Diophantine problems in solvable groups.Bull. Math. Sci., 10(1):2050005, 27, 2020

    2020

  51. [51]

    The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth

    F. Gautero and M. Lustig. The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth.arXiv preprint arXiv:0707.0822, 2007. 30 UNDECIDABLE DIOPHANTINE PROBLEMS

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  52. [52]

    P. Ghosh. Relative hyperbolicity of free-by-cyclic extensions.Compos. Math., 159(1):153–183, 2023

    2023

  53. [53]

    R. D. Gray and A. Levine. Decidability of equations and first-order theory in Seifert 3-manifold groups. arXiv preprint arXiv:2512.07690, 2025

    work page arXiv 2025

  54. [54]

    M. Gromov. Hyperbolic groups. InEssays in group theory, volume 8 ofMath. Sci. Res. Inst. Publ., pages 75–263. Springer, New York, 1987

    1987

  55. [55]

    J. Howie. On pairs of 2-complexes and systems of equations over groups.J. Reine Angew. Math., 324:165– 174, 1981

    1981

  56. [56]

    J. Howie. Cohomology of one-relator products of locally indicable groups.J. London Math. Soc. (2), 30(3):419–430, 1984

    1984

  57. [57]

    J. Howie. The quotient of a free product of groups by a single high-powered relator. II. Fourth powers.Proc. London Math. Soc. (3), 61(1):33–62, 1990

    1990

  58. [58]

    Jaikin-Zapirain, M

    A. Jaikin-Zapirain, M. Linton, and P. S´ anchez-Peralta. Group pairs, coherence and Farrell–Jones conjecture forK 0.arXiv preprint arXiv:2510.23518, 2025

    work page arXiv 2025

  59. [59]

    Kharlampovich and L

    O. Kharlampovich and L. L´ opez. Bi-interpretability of some monoids with the arithmetic and applications. Semigroup Forum, 99(1):126–139, 2019

    2019

  60. [60]

    Kharlampovich, L

    O. Kharlampovich, L. L´ opez, and A. Myasnikov. The Diophantine problem in some metabelian groups. Mathematics of computation, 89(325):2507–2519. Corrected version: arxiv.org/abs/1903.10068, 2020

    work page arXiv 1903

  61. [61]

    Kharlampovich and A

    O. Kharlampovich and A. Miasnikov. The Diophantine problem in iterated wreath products of free abelian groups is undecidable.arXiv e-prints, arXiv:2502.09442v1, 2025

    work page arXiv 2025

  62. [62]

    Koymans and C

    P. Koymans and C. Pagano. Hilbert’s tenth problem via additive combinatorics.arXiv preprint arXiv:2412.01768, 2024

    work page arXiv 2024

  63. [63]

    P. H. Kropholler. Baumslag-Solitar groups and some other groups of cohomological dimension two.Com- mentarii Mathematici Helvetici, 65(1):547–558, 1990

    1990

  64. [64]

    P. H. Kropholler. A group theoretic proof of the torus theorem.Geometric group theory, 1:138–158, 1993

    1993

  65. [65]

    Kudlinska

    M. Kudlinska. On subgroup separability of free-by-cyclic and deficiency 1 groups.Bulletin of the London Mathematical Society, 56(1):338–351, 2024

    2024

  66. [66]

    A. Levine. Equations in virtually class 2 nilpotent groups.J. Groups Complex. Cryptol., 14(1):Paper No. 2, 17, 2022

    2022

  67. [67]

    G. Levitt. Generalized Baumslag–Solitar groups: rank and finite index subgroups. InAnnales de l’Institut Fourier, volume 65, pages 725–762, 2015

    2015

  68. [68]

    M. Linton. The geometry of subgroups of mapping tori of free groups.arXiv preprint arXiv:2510.03145, 2025

    work page arXiv 2025

  69. [69]

    Linton and C.-F

    M. Linton and C.-F. Nyberg-Brodda. The theory of one-relator groups: history and recent progress.arXiv preprint arXiv:2501.18306, 2025

    work page arXiv 2025

  70. [70]

    Lipschutz

    S. Lipschutz. Generalization of Dehn’s result on the conjugacy problem.Proc. Amer. Math. Soc., 17:759–762, 1966

    1966

  71. [71]

    A. D. Logan. The conjugacy problem for ascending HNN-extensions of free groups.Proc. Lond. Math. Soc. (3), 131(4):Paper No. e70088, 24, 2025

    2025

  72. [72]

    Lohrey and G

    M. Lohrey and G. S´ enizergues. Theories of HNN-extensions and amalgamated products. InAutomata, languages and programming. Part II, volume 4052 ofLecture Notes in Comput. Sci., pages 504–515. Springer, Berlin, 2006

    2006

  73. [73]

    R. C. Lyndon, P. E. Schupp, R. Lyndon, and P. Schupp.Combinatorial group theory, volume 89. Springer, 1977

    1977

  74. [74]

    Magnus, A

    W. Magnus, A. Karrass, and D. Solitar.Combinatorial group theory: Presentations of groups in terms of generators and relations. Courier Corporation, 2004

    2004

  75. [75]

    G. S. Makanin. The problem of the solvability of equations in a free semigroup.Mat. Sb. (N.S.), 103(145)(2):147–236, 319, 1977

    1977

  76. [76]

    G. S. Makanin. Equations in a free group.Izv. Akad. Nauk SSSR Ser. Mat., 46(6):1199–1273, 1344, 1982

    1982

  77. [77]

    Mandel and A

    R. Mandel and A. Ushakov. The Diophantine problem for systems of algebraic equations with exponents. J. Algebra, 636:779–803, 2023

    2023

  78. [78]

    Mandel and A

    R. Mandel and A. Ushakov. Quadratic equations in metabelian Baumslag-Solitar groups.Internat. J. Algebra Comput., 33(6):1195–1216, 2023

    2023

  79. [79]

    Martino and E

    A. Martino and E. Ventura. A description of auto-fixed subgroups in a free group.Topology, 43(5):1133–1164, 2004

    2004

  80. [80]

    J. V. Matijaseviˇ c. The Diophantineness of enumerable sets.Dokl. Akad. Nauk SSSR, 191:279–282, 1970

    1970

Showing first 80 references.