Spanning trees in hyperbolic graphs

We construct spanning trees in locally finite hyperbolic graphs that represent their hyperbolic compactification in a good way: so that the tree has at least one but at most a bounded number of disjoint rays to each boundary point. As a corollary we extend a result of Gromov which says that from every hyperbolic graph with bounded degrees one can construct a tree (disjoint from the graph) with a continuous surjection from the ends of the tree onto the hyperbolic boundary such that the surjection is finite-to-one. We shall construct a tree with these properties as a subgraph of the hyperbolic graph, which in addition is also a spanning tree of that graph.     

Pointed spherical tilings and hyperbolic virtual polytopes

The paper presents a shortcut introduction to the theory of hyperbolic virtual polytopes from the point of view of combinatorial rigidity. (It is assumed that the reader is acquainted with the notions of Laman graph, 3D lifting, and pointed tiling.) From this point of view, a hyperbolic virtual polytope is a stressed pointed graph embedded in the sphere S ². The advantage of such a presentation is that it gives an alternative and most convincing proof of existence of hyperbolic virtual polytopes. Bibliography: 20 titles.     

A full characterization of invariant embeddability of unimodular planar graphs

When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry-invariant? This question was answered for one-ended unimodular graphs in Benjamini and Timar, using the fact that such graphs automatically have locally finite (simply connected) drawings into the plane. For the case of graphs with multiple ends the question was left open. We revisit Halin's graph theoretic characterization of graphs that have a locally finite embedding into the plane. Then we prove that such unimodular random graphs do have a locally finite invariant embedding into the Euclidean or the hyperbolic plane, depending on whether the graph is amenable or not.     

Some geometrical aspects of a maximal three-coloured triangle-free graph

A three-coloured triangle-free complete graph with 16 vertices is constructed in an ad hoc manner. The edges of one colour in the complete graph, with the 16 vertices, form a Greenwood-Gleason graph, which can be regarded as the edges and diagonals of a hypercube in four dimensions, and which also has a representation as a graph in five dimensions all of whose automorphisms are isometries. In the complete graph, the blue edges form 40 quadrilaterals; 20 of these have red diagonals, and these 20 “red quadrilaterals,” meeting along 40 edges and at 16 vertices, represent a topological surface of characteristic ?4, a Klein bottle with two handles. This surface can be represented using a tessellation of regular quadrilaterals in the hyperbolic plane. To obtain the only other three-coloured triangle-free complete graph with 16 vertices some of the blue and red edges are interchanged in a way that can be described very simply using either the surface of characteristic ?4 or the hyperbolic tessellation.     

Expanders are not hyperbolic

We bound from above the number of vertices of a graph in terms of the Cheeger constant and the δ-hyperbolicity of the graph. As a corollary we get that expanders are not uniformly hyperbolic.     

Minimisers of the Allen–Cahn equation on hyperbolic graphs

We investigate minimal solutions of the Allen–Cahn equation on a Gromov-hyperbolic graph. Under some natural conditions on the graph, we show the existence of non-constant uniformly-bounded minimal solutions with prescribed asymptotic behaviours. For a phase field model on a hyperbolic graph, such solutions describe energy-minimising steady-state phase transitions that converge towards prescribed phases given by the asymptotic directions on the graph.     

Height in splittings of relatively hyperbolic groups

Geometriae Dedicata - Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex...     

Singular hyperbolicity of star flows on surfaces

In this paper, we prove that every star flow on the closed surface has finitely many chain recurrent classes. Furthermore, it is singular hyperbolic if every non-trivial singular chain component is a graph. As a consequence, every star flow on the 2-sphere or the projective plane is singular hyperbolic.     

A Combination Theorem for Metric Bundles

We introduce the notion of metric (graph) bundles which provide a coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina?CFeighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the existence of quasi-isometric sections in this generality. Then we prove a combination theorem for metric (graph) bundles that establishes sufficient conditions, particularly flaring, under which the metric bundles are hyperbolic. We use this to give examples of surface bundles over hyperbolic disks, whose universal cover is Gromov-hyperbolic. We also show that in typical situations, flaring is also a necessary condition.     

Matching arc complexes: Connectedness and hyperbolicity

Addressing a question of Zaremsky, we give conditions on a finite simplicial graph which guarantee that the associated matching arc complex is connected and hyperbolic.     

On minimal graph evolutions in the hyperbolic space

We study here minmal graph evolutions in the hyperbolic space and prove that there exists a unique smooth solution. This work is partially supported by NNSF of China     

Cubic vertex-transitive graphs of girth six

In this paper, a complete classification of finite simple cubic vertex-transitive graphs of girth 6 is obtained. It is proved that every such graph, with the exception of the Desargues graph on 20 vertices, is either a skeleton of a hexagonal tiling of the torus, the skeleton of the truncation of an arc-transitive triangulation of a closed hyperbolic surface, or the truncation of a 6-regular graph with respect to an arc-transitive dihedral scheme. Cubic vertex-transitive graphs of girth larger than 6 are also discussed.     

Three-component WKI equation and curve motion flow in Euclidean and Minkowski space

Motion of curves in the four-dimensional Euclidean and Minkowski space are discussed. It is shown that the three-component WKI equation and its hyperbolic type arise from certain curve motion flows. They are obtained by using the relation between curvatures of the curves and their graph. Group-invariant solutions to the three-component WKI equation and its hyperbolic type are also derived.     

Circle boundaries of planar graphs

Letg be an infinite, connected, planar graph with bounded vertex degree, which obeys a strong isoperimetric inequality and which can be embedded in the plane so that each cycle surrounds only finitely many vertices. We investigate a certain class of compactifications ofg; one of which has boundary homemorophic to a circle. We shall show that ifg is a tree or, more generally, ifg is hyperbolic, then this circle boundary supports an integral representation of any given bounded harmonic function. We further show that in the specific case of a triangulation of the plane, the graph is hyperbolic and therefore the Martin boundary is a circle.     

Dihedral Branches Coverings of Hyperbolic Orbifolds

We consider the class of hyperbolic 3-orbifolds whose underlying topological space is the 3-sphere S ³ and whose singular set is a trivalent graph with singular index 2 along each edge (an important special case occurs when the trivalent graph is the 1-skeleton of a hyperbolic polyhedron). Our main result is a classification of the D-branched coverings of these orbifolds (where D₂ is the dihedral group of order 4) under some general conditions on their isometry groups or the lengths of their geodesics.     

Embedding right-angled Artin groups into graph braid groups

We construct an embedding of any right-angled Artin group G(Δ) defined by a graph Δ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of Δ. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.      

On the hyperbolic unitary geometry

Hans Cuypers (Preprint) describes a characterisation of the geometry on singular points and hyperbolic lines of a finite unitary space—the hyperbolic unitary geometry—using information about the planes. In the present article we describe an alternative local characterisation based on Cuypers’ work and on a local recognition of the graph of hyperbolic lines with perpendicularity as adjacency. This paper can be viewed as the unitary analogue of the second author’s article (J. Comb. Theory Ser. A 105:97–110, 2004) on the hyperbolic symplectic geometry.     

Approximation properties of fine hyperbolic graphs

In this paper, we propose a definition of approximation property which is called the metric invariant translation approximation property for a countable discrete metric space. Moreover, we use the techniques of Ozawa’s to prove that a fine hyperbolic graph has the metric invariant translation approximation property.     

WEIERSTRASS REPRESENTATION FORSURFACES WITH PRESCRIBED NORMALGAUSS MAP AND GAUSS CURVATURE IN H~3

The author obtains a Weierstrass representation for surfaces with prescribed normal Gauss map and Gauss curvature in H3. A differential equation about the hyperbolic Gauss map is also obtained, which characterizes the relation among the hyperbolic Gauss map, the normal Gauss map and Gauss curvature. The author discusses the harmonicity of the normal Gauss map and the hyperbolic Gauss map from surface with constant Gauss curvature in H3 to S2 with certain altered conformal metric. Finally, the author considers the surface whose normal Gauss map is conformal and derives a completely nonlinear differential equation of second order which graph must satisfy.     

When is the graph product of hyperbolic groups hyperbolic?

Given a finite simplicial graph , and an assignment of groups to the verticles of , the graph product is the free product of the vertex groups modulo relations implying that adjacent vertex groups commute. We use Gromov's link criteria for cubical complexes and techniques of Davis and Moussang to study the curvature of graph products of groups. By constructing a CAT(–1) cubical complex, it is shown that the graph product of word hyperbolic groups is itself word hyperbolic if and only if the full subgraph in , generated by vertices whose associated groups are finite, satisfies three specific criteria. The construction shows that arbitrary graph products of finite groups are Bridson groups.