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A well‐known theorem of Whitney states that a 3‐connected planar graph admits an essentially unique embedding into the 2‐sphere. We prove a 3‐dimensional analogue: a simply‐connected 2‐complex every link graph of which is 3‐connected admits an essentially unique locally flat embedding into the 3‐sphere, if it admits one at all. This can be thought of as a generalisation of the 3‐dimensional Schoenflies theorem. [ABSTRACT FROM AUTHOR]

We introduce a strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, in particular, the existence of Dirichlet harmonic function s proved by Benjamini and Schramm. As a corollary we obtain that every planar nonamenable graph admits nonconstant Dirichlet harmonic function s27. [ABSTRACT FROM AUTHOR]

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 construct minor-closed addable families of graphs that are subcritical and contain all planar graphs. This contradicts (one direction of) a well-known conjecture of Noy. [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]

The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps. [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]