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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Lohrey, Markus

CiteScore 2018: 0.80

SCImago Journal Rank (SJR) 2018: 0.368
Source Normalized Impact per Paper (SNIP) 2018: 1.061

Mathematical Citation Quotient (MCQ) 2018: 0.38

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On finitely generated submonoids of virtually free groups

Pedro V. Silva / Alexander Zakharov
Published Online: 2018-10-17 | DOI: https://doi.org/10.1515/gcc-2018-0008


We prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization of graded submonoids in virtually free groups as quasi-geodesic submonoids, and show that their word problem is rational (as a relation). We also solve the isomorphism problem for this class of monoids, generalizing earlier results for submonoids of free monoids. We also prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups.

Keywords: Free group; virtually free group; submonoids; graded monoid; rational monoid; Kleene monoids; decidability

MSC 2010: 20E05; 20M05; 20F10; 68Q45; 68Q70


  • [1]

    Y. Antolín, On Cayley graphs of virtually free groups, Groups Complex. Cryptol. 3 (2011), no. 2, 301–327. Google Scholar

  • [2]

    V. Araújo and P. V. Silva, Geometric characterizations of virtually free groups, J. Algebra Appl. 16 (2017), no. 9, Article ID 1750180. Web of ScienceGoogle Scholar

  • [3]

    L. Bartholdi and P. V. Silva, Rational subsets of groups, preprint (2010), https://arxiv.org/abs/1012.1532; Chapter 23 of the Handbook AutoMathA (to appear).

  • [4]

    J. Berstel, Transductions and Context-free Languages, Leitfäden Angew. Math. Mech. 38, B. G. Teubner, Stuttgart, 1979. Google Scholar

  • [5]

    M. R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Grundlehren Math. Wiss. 319, Springer, Berlin, 1999. Google Scholar

  • [6]

    A. J. Cain, E. F. Robertson and N. Ruškuc, Subsemigroups of virtually free groups: finite Malcev presentations and testing for freeness, Math. Proc. Cambridge Philos. Soc. 141 (2006), no. 1, 57–66. CrossrefGoogle Scholar

  • [7]

    C. Choffrut, T. Harju and J. Karhumäki, A note on decidability questions on presentations of word semigroups, Theoret. Comput. Sci. 183 (1997), no. 1, 83–92. CrossrefGoogle Scholar

  • [8]

    V. Diekert and A. Weiß, Context-free groups and Bass-Serre theory, Algorithmic and Geometric Topics Around Free Groups and Automorphisms, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham (2017), 43–110. Google Scholar

  • [9]

    Z. Grunschlag, Algorithms in Geometric Group Theory, ProQuest LLC, Ann Arbor, 1999; Thesis (Ph.D.)–University of California, Berkeley. Google Scholar

  • [10]

    J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading, 1979. Google Scholar

  • [11]

    J. H. Johnson, Do rational equivalence relations have regular cross sections?, Automata, Languages and Programming (Nafplion 1985), Lecture Notes in Comput. Sci. 194, Springer, Berlin (1985), 300–309. Google Scholar

  • [12]

    J. H. Johnson, Rational equivalence relations, Theoret. Comput. Sci. 47 (1986), no. 1, 39–60. CrossrefGoogle Scholar

  • [13]

    I. Kapovich and A. Myasnikov, Stallings foldings and subgroups of free groups, J. Algebra 248 (2002), no. 2, 608–668. CrossrefGoogle Scholar

  • [14]

    S. W. Margolis, J. Meakin and Z. Šuniḱ, Distortion functions and the membership problem for submonoids of groups and monoids, Geometric Methods in Group Theory, Contemp. Math. 372, American Mathematical Society, Providence (2005), 109–129. Google Scholar

  • [15]

    K. A. Mihaĭlova, The occurrence problem for direct products of groups, Dokl. Akad. Nauk SSSR 119 (1958), 1103–1105. Google Scholar

  • [16]

    D. E. Muller and P. E. A. Schupp, Groups, the theory of ends, and context-free languages, J. Comput. System Sci. 26 (1983), no. 3, 295–310. CrossrefGoogle Scholar

  • [17]

    M. Neunhöffer, M. Pfeiffer and N. Ruškuc, Deciding word problems of semigroups using finite state automata, preprint (2012), https://arxiv.org/abs/1206.1714.

  • [18]

    M. Pelletier and J. Sakarovitch, Easy multiplications. II. Extensions of rational semigroups, Inform. and Comput. 88 (1990), no. 1, 18–59. CrossrefGoogle Scholar

  • [19]

    C. P. Rupert, On commutative Kleene monoids, Semigroup Forum 43 (1991), no. 2, 163–177. Web of ScienceCrossrefGoogle Scholar

  • [20]

    J. Sakarovitch, Easy multiplications. I. The realm of Kleene’s theorem, Inform. and Comput. 74 (1987), no. 3, 173–197. CrossrefGoogle Scholar

  • [21]

    J. Sakarovitch, Elements of Automata Theory, Cambridge University Press, Cambridge, 2009. Google Scholar

  • [22]

    P. V. Silva, Recognizable subsets of a group: finite extensions and the abelian case, Bull. Eur. Assoc. Theor. Comput. Sci. EATCS (2002), no. 77, 195–215. Google Scholar

  • [23]

    P. V. Silva, The homomorphism problem for the free monoid, Discrete Math. 259 (2002), no. 1–3, 189–200. CrossrefGoogle Scholar

  • [24]

    J. R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88 (1968), 312–334. CrossrefGoogle Scholar

  • [25]

    J. R. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), no. 3, 551–565. CrossrefGoogle Scholar

About the article

Received: 2017-12-21

Published Online: 2018-10-17

Published in Print: 2018-11-01

Funding Source: Centro de Matemática Universidade do Porto

Award identifier / Grant number: UID/MAT/00144/2013

Funding Source: H2020 European Research Council

Award identifier / Grant number: 336983

Funding Source: Eusko Jaurlaritza

Award identifier / Grant number: IT974-16

Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2014-53810-C2-2-P

Funding Source: Russian Foundation for Basic Research

Award identifier / Grant number: 15-01-05823

Both authors were partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER), under the partnership agreement PT2020. The second author was also partially supported by the ERC Grant 336983, by the Basque Government grant IT974-16, by the grant MTM2014-53810-C2-2-P of the Ministerio de Economía y Competitividad of Spain, and by the Russian Foundation for Basic Research (project no. 15-01-05823).

Citation Information: Groups Complexity Cryptology, Volume 10, Issue 2, Pages 63–82, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2018-0008.

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