Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

IMPACT FACTOR 2018: 0.490

CiteScore 2018: 0.47

SCImago Journal Rank (SJR) 2018: 0.279
Source Normalized Impact per Paper (SNIP) 2018: 0.627

Mathematical Citation Quotient (MCQ) 2017: 0.26

See all formats and pricing
More options …
Volume 66, Issue 3


On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups

Giovanni Moreno / Monika Ewa Stypa
Published Online: 2016-08-17 | DOI: https://doi.org/10.1515/ms-2015-0154


We prove that the Cayley graph and the coset geometry of the von Dyck group D(a, b, c) are linked by a vertex-to-edge duality.

MSC 2010: Primary 05E18; 20F65; 05C25; 05C15; 52C20; 51E99; 20F05; 22E40

Keywords: group actions on combinatorial structures; geometric group theory; incidence geometry; graphs and abstract algebra; tilings in 2 dimensions

The first author was supported by the project P201/12/G028 of the Czech Republic Grant Agency (GA ČR).

The second author was supported by the doctoral school of the University of Salerno.


  • [1]

    Beardon, A. F.: The Geometry of Discrete Groups. Grad. Texts in Math. 91, Springer-Verlag, New York, 1995.Google Scholar

  • [2]

    Beineke, L. W.: Characterizations of derived graphs, J. Combin. Theory 9 (1970), 129–135.Google Scholar

  • [3]

    Conder, M. D. E.—Martin, G. J.: Cusps, triangle groups and hyperbolic 3-folds, J. Austral. Math. Soc. Ser. A Math. Statist. 55 (1993), 149–182.Google Scholar

  • [4]

    Coxeter, H. S. M.—Moser, W. O. J.: Generators and Relations for Discrete Groups (3rd ed.). Ergeb. Math. Grenzgeb. 14, Springer-Verlag, New York, 1972.Google Scholar

  • [5]

    Dyck, W.: Gruppentheoretische Studien, Math. Ann. 20 (1882), 1–44. http://dx.doi.org/10.1007/BF01443322

  • [6]

    Fazio, N.—Iga, K.—Nicolosi, A.—Perret, L.—Skeith III, W. E.: Hardness of Learning Problems over Burnside Groups of Exponent 3. Cryptology ePrint Archive, Report 2011/398, 2011. http://eprint.iacr.org/2011/398

  • [7]

    Giudici, M.—Pearce, G.—Praeger, Ch. E.: Basic coset geometries, J. Algebraic Combin. 36 (2012), 561–594. http://dx.doi.org/10.1007/s10801-012-0350-8

  • [8]

    Gross, J. L.—Yellen, J.: Graph Theory and Its Applications 2nd ed.. Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2006.Google Scholar

  • [9]

    GrÜnbaum, B.—Shephard, G. C.: Tilings and Patterns, W. H. Freeman and Company, New York, 1989.Google Scholar

  • [10]

    Gupta, N.: On groups in which every element has finite order, Amer. Math. Monthly 96 (1989), 297–308. http://dx.doi.org/10.2307/2324085

  • [11]

    Handbook of Incidence Geometry F. Buekenhout, ed.). Buildings and Foundations, North-Holland, Amsterdam, 1995.Google Scholar

  • [12]

    Magnus, W.—Karrass, A.—Solitar, D.: Combinatorial Group Theory (2nd ed.), Dover Publications Inc., Mineola, NY, 2004.Google Scholar

  • [13]

    Mckee, T. A.—McMorris, F. R.: Topics in Intersection Graph Theory, SIAM Monogr. Discrete Math. Appl., Soc. for Industrial and Appl. Math. (SIAM), Philadelphia, PA, 1999. http://dx.doi.org/10.1137/1.9780898719802

  • [14]

    Neumann, P. M.: The SQ-universality of some finitely presented groups, J. Austral. Math. Soc. Math. Statist. 16 (1973), 1–6 (Collection of articles dedicated to the memory of Hanna Neumann, I).Google Scholar

  • [15]

    Pasini, A.: Geometries and chamber systems, Sūrikaisekikenkyūsho Kōkyūroku 867 (1994), 28–52.Google Scholar

  • [16]

    Peters, J.—Naimpally, S.: Applications of near sets, Notices Amer. Math. Soc. 59 (2012), 536–542. http://dx.doi.org/10.1090/noti817

  • [17]

    Robinson, D. J. S.: A course in the theory of groups (2nd ed.). Grad. Texts in Math. 80, Springer-Verlag, New York, 1996. http://dx.doi.org/10.1007/978-1-4419-8594-1

  • [18]

    Ronan, M. A.: Triangle geometries, J. Combin. Theory Ser. A 37 (1984), 294–319. http://dx.doi.org/10.1016/0097-3165(84)90051-7

  • [19]

    Sabidussi, G.: On a class of fixed-point-free graphs, Proc. Amer. Math. Soc. 9 (1958), 800–804.Google Scholar

  • [20]

    Seward, B.: Burnside’s problem, spanning trees, and tilings, arXiv (2011), http://arxiv.org/abs/1104.1231v2

  • [21]

    Sossinsky, A. B.: Tolerance space theory and some applications, Acta Appl. Math. 5 (1986), 137–167. http://dx.doi.org/10.1007/BF00046585

  • [22]

    Tucker, T. W.: Finite groups acting on surfaces and the genus of a group, J. Combin. Theory Ser. B 34 (1983), 82–98. http://dx.doi.org/10.1016/0095-8956(83)90009-6

  • [23]

    Vinogradov, A. M.: Why is space three-dimensional and how may groups be seen?, Acta Appl. Math. 5 (1986), 169–180. http://dx.doi.org/10.1007/BF00046586

  • [24]

    White, A. T.: On the genus of a group, Trans. Amer. Math. Soc. 173 (1972), 203–214.Google Scholar

  • [25]

    Zeeman, E. C.: The topology of the brain and visual perception. In: Topology of 3-manifolds and Related Topics. Proc. The Univ. of Georgia Institute, 1961, Prentice-Hall, Englewood Cliffs, NJ, 1962, pp. 240–256.Google Scholar

  • [26]

    Tits: Géométries polyédriques et groupes simples, U Atti 2a Riunione Groupem. Math. Express. Lat. Firenze (1962), 66–88.Google Scholar

About the article

Received: 2013-07-12

Accepted: 2013-10-23

Published Online: 2016-08-17

Published in Print: 2016-06-01

Citation Information: Mathematica Slovaca, Volume 66, Issue 3, Pages 527–538, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2015-0154.

Export Citation

© 2016 Mathematical Institute Slovak Academy of Sciences.Get Permission

Comments (0)

Please log in or register to comment.
Log in