Gromov hyperbolic tessellation graphs

If X is a geodesic metric space and x(1), x(2), x(3) is an element of X, a geodesic triangle T with vertices x(1), x(2), x(3), is the union of three geodesics [x(1)x(2)], [x(2)x(3)] and [x(3)x(1)] in X. The space X is delta-hyperbolic (in the Gromov sense) if any side of T is contained in a delta-neighborhood of the union of the two other sides, for every geodesic triangle T in X. In this paper we obtain criteria which allow us to decide for a large class of graphs whether they are hyperbolic or not. We also obtain results about tessellations of general Riemannian surfaces with a lower bound for the curvature. Surprisingly, these results on Riemannian surfaces are the key in order to obtain further information about tessellations of the Euclidean plane, and they even allow to answer a fundamental question: how do changes in the lengths of the edges of a general graph influence on the hyperbolicity? We also prove that a tessellation graph is hyperbolic if and only if its dual graph is hyperbolic.

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