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Let G be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface X ⊆ S2. We prove that G admits such an action that is in addition co-compact, provided we can replace X by another surface Y ⊆ S2. We also prove that if a group H has a finitely generated Cayley (multi-) graph C equivariantly embeddable in S2, then C can be chosen so as to have no infinite path on the boundary of a face. The proofs of these facts are intertwined, and the classes of groups they define coincide. In the orientation-preserving case they are exactly the (isomorphism types of) finitely generated Kleinian function groups. We construct a finitely generated planar Cayley graph whose group is not in this class. In passing, we observe that the Freudenthal compactification of every planar surface is homeomorphic to the sphere. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We show that a group admits a planar, finitely generated, Cayley graph if and only if it admits a special kind of group presentation we introduce, called a planar presentation. Planar presentations can be recognised algorithmically. As a consequence, we obtain an effective enumeration of the planar Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al. asking whether the planar groups can be effectively enumerated. [ABSTRACT FROM AUTHOR]

We prove that every vertex transitive, planar, 1-ended, graph covers every graph whose balls of radius r are isomorphic to the ball of radius r in G for a sufficiently large r . It is natural to ask whether this is true for all finitely presented Cayley graphs, but the answer turns out to be negative (de la Salle and Tessera, 2015). [ABSTRACT FROM AUTHOR]

We classify the planar cubic Cayley graphs of connectivity 2, providing an explicit presentation and embedding for each of them. Combined with Georgakopoulos (2017) this yields a complete description of all planar cubic Cayley graphs. [ABSTRACT FROM AUTHOR]

Abstract: We prove that a Cayley graph can be embedded in the Euclidean plane without accumulation points of vertices if and only if it is the 1-skeleton of a Cayley complex that can be embedded in the plane after removing redundant simplices. We also give a characterisation of these Cayley graphs in term of group presentations, and deduce that they can be effectively enumerated. [Copyright &y& Elsevier]

A well-known result of Benjamini, Lyons, Peres, and Schramm states that if G is a finitely generated Cayley graph of a group Γ , then Γ is amenable if and only if G admits a Γ -invariant random spanning tree with at most two ends. We show that this is equivalent to the existence of a Γ -invariant random spanning double ray in a power of G. [ABSTRACT FROM AUTHOR]