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The Planar Cayley Graphs are Effectively Enumerable I: Consistently Planar Graphs

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Abstract

We obtain an effective enumeration of the family of finitely generated groups admitting a faithful, properly discontinuous action on some 2-manifold contained in the sphere. This is achieved by introducing a type of group presentation capturing exactly these groups.

Extending this in a companion paper, we find group presentations capturing the planar finitely generated Cayley graphs. Thus we obtain an effective enumeration of these Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al.

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References

  1. S. I. Adjan: The algorithmic unsolvability of checking certain properties of groups, Dokl. Akad. Nauk SSSR103 (1955), 533–535.

    MathSciNet  Google Scholar 

  2. G. N. Arzhantseva and P.-A. Cherix: On the Cayley graph of a generic finitely presented group, Bulletin of the Belgian Mathematical Society–Simon Stevin11 (2004), 589–601.

    MathSciNet  MATH  Google Scholar 

  3. G. Baumslag: Topics in Combinatorial Group Theory, Birkhauser Verlag, Basel, 1993.

    Book  Google Scholar 

  4. O. Bogopolski: Introduction to Group Theory, EMS, Zuerich, Switzerland, 2008.

    Book  Google Scholar 

  5. W. Dicks and M. J. Dunwoody: Groups acting on graphs, Cambridge University Press, 1989.

    MATH  Google Scholar 

  6. R. Diestel: Graph Theory (3rd edition), Springer-Verlag, 2005.

    MATH  Google Scholar 

  7. C. Droms: Infinite-ended groups with planar Cayley graphs, J. Group Theory9 (2006), 487–496.

    Article  MathSciNet  Google Scholar 

  8. C. Droms, B. Servatius and H. Servatius: Connectivity and planarity of Cayley graphs, Beitr. Algebra Geom.39 (1998), 269–282.

    MathSciNet  MATH  Google Scholar 

  9. M. J. Dunwoody: Planar graphs and covers, Preprint.

  10. M. J. Dunwoody: The accessibility of finitely presented groups, Invent. Math.81 (1985), 449–457.

    Article  MathSciNet  Google Scholar 

  11. A. Georgakopoulos: Characterising planar Cayley graphs and Cayley complexes in terms of group presentations, Europ. J. Comb.36 (2014), 282–293.

    Article  MathSciNet  Google Scholar 

  12. A. Georgakopoulos: The planar cubic Cayley graphs, Memoirs of the AMS, 250 (1190), 2017.

    Google Scholar 

  13. A. Georgakopoulos: The planar cubic Cayley graphs of connectivity 2, Europ. J. Comb.64 (2017), 152–169.

    Article  MathSciNet  Google Scholar 

  14. A. Georgakopoulos and M. Hamann: The planar Cayley graphs are effectively enumerable II, In preparation. (Older version available as part of arXiv: 1506.03361).

  15. M. Hamann: Planar transitive graphs, submitted, arXiv:1511.08777.

  16. M. Hamann: Generating the cycle space of planar graphs, Electron. J. of Comb.22 (2015), 2.34.

    MathSciNet  MATH  Google Scholar 

  17. W. Imrich: On Whitney’s theorem on the unique embeddability of 3-connected planar graphs, in: Recent Adv. Graph Theory, Proc. Symp. Prague 1974, 303–306, 1975.

    Google Scholar 

  18. R. C. Lyndon and P. E. Schupp: Combinatorial Group Theory, Springer Science & Business Media, January 2001.

    Book  Google Scholar 

  19. A. M. Macbeath: The classification of non-euclidean plane crystallographic groups, Can. J.Math.19 (1967), 1192–1205.

    Article  MathSciNet  Google Scholar 

  20. H. Maschke: The representation of finite groups, especially of rotation groups of three and four dimensional space, by Cayley’s color diagrams, Amer. J. Math18 (1896), 156–194.

    Article  MathSciNet  Google Scholar 

  21. J. Meier:Groups, Graphs and Trees, Cambridge University Press, 2008.

    Book  Google Scholar 

  22. B. Mohar: Tree amalgamation of graphs and tessellations of the cantor sphere, J. Combin. Theory (Series B)96 (2006), 740–753.

    Article  MathSciNet  Google Scholar 

  23. B. Fine, P. Ackermann and G. Rosenberger: On surface groups: Motivating examples in combinatorial group theory, in: Groups St Andrews 2005, 96–129, Cambridge University Press, 2007.

    MATH  Google Scholar 

  24. H. Poincaré: Theorie des groupes fuchsiens, Acta Mathematica1 (1882), 1–62.

    Article  MathSciNet  Google Scholar 

  25. M. O. Rabin: Recursive unsolvability of group theoretic problems, Annals of Mathematics. 67 (1958), 172–194.

    Article  MathSciNet  Google Scholar 

  26. T. W. Tucker: Finite Groups Acting on Surfaces and the Genus of a Group, J. Combin. Theory (Series B) 34 (1983), 82–98.

    Article  MathSciNet  Google Scholar 

  27. H. Whitney: Congruent graphs and the connectivity of graphs, American J. of Mathematics54 (1932), 150–168.

    Article  MathSciNet  Google Scholar 

  28. H. C. Wilkie: On non-Euclidean crystallographic groups, Math. Z.91 (1965), 87–102.

    Article  MathSciNet  Google Scholar 

  29. H. Zieschang, E. Vogt and H.-D. Coldewey: Surfaces and planar discontinuous groups, Revised and expanded transl. from the German by J. Stillwell. Lecture Notes in Mathematics 835. Springer-Verlag, 1980.

    Book  Google Scholar 

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Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Warwick, CV4 7AL, UK

    Agelos Georgakopoulos

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    Matthias Hamann

Authors
  1. Agelos Georgakopoulos
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  2. Matthias Hamann
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Additional information

Supported by EPSRC grant EP/L002787/1, and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 639046). The first author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme ‘Random Geometry’ where work on this paper was undertaken.

Both authors have been supported by FWF grant P-19115-N18.

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Georgakopoulos, A., Hamann, M. The Planar Cayley Graphs are Effectively Enumerable I: Consistently Planar Graphs. Combinatorica 39, 993–1019 (2019). https://doi.org/10.1007/s00493-019-3763-3

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  • DOI: https://doi.org/10.1007/s00493-019-3763-3

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