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We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex-transitive graph. [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]

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

Answering a question of Benjamini and Schramm (Ann Probab 24(3):1219-1238, ), we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, which arises from a discrete version of Riemann's mapping theorem. This implies a conjecture of Northshield (Potential Anal 2(4):299-314, ). Some of our technique apply to the non-planar case and might have further applications. When the graph is also hyperbolic then, under mild conditions, we prove the equivalence of several boundary constructions. [ABSTRACT FROM AUTHOR]

We show that there is no one-ended, locally finite, planar, hyperbolic graph such that the stabilizer of one of its hyperbolic boundary points acts transitively on the vertices of the graph. This gives a partial answer to a question by Kaimanovich and Woess. [ABSTRACT FROM AUTHOR]

Eberhard proved that for every sequence ( p), 3≤ k≤ r, k≠6, of nonnegative integers satisfying Euler's formula ∑(6− k) p=12, there are infinitely many values p such that there exists a simple convex polyhedron having precisely p faces of size k for every k≥3, where p=0 if k> r. In this paper we prove a similar statement when nonnegative integers p are given for 3≤ k≤ r, except for k=5 and k=7 (but including p). We prove that there are infinitely many values p, p such that there exists a simple convex polyhedron having precisely p faces of size k for every k≥3. We derive an extension to arbitrary closed surfaces, yielding maps of arbitrarily high face-width. Our proof suggests a general method for obtaining results of this kind. [ABSTRACT FROM AUTHOR]

We prove that every rayless graph has an unfriendly partition. [ABSTRACT FROM AUTHOR]