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


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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Corresponding author

Correspondence to Agelos Georgakopoulos.

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).

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Mathematics Subject Classification (2010)

  • 05C25
  • 05C10
  • 20F05
  • 20F65