Speaker: Dr. Walter Carballosa

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Venue:TBA

Dr. Walter Carballosa, (MDC)

Title: On the Gromov hyperbolicity of planar graphs.

Abstract: The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this work we obtain criteria which allow us to decide for a large class of graphs whether they are hyperbolic or not: our main interest are the planar graphs which are the ``boundary" (the $1$-skeleton) of a tessellation of the Euclidean plane; however, 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 consider the conjecture which states that every tessellation graph of $\mathbb{R}^2$ with convex tiles is non-hyperbolic. However, we show that this conjecture is false. But, this fact depends strongly of the specific distribution of vertices set and the choice of edges.