Hyperbolic polyhedral surfaces with regular faces

We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least $2\pi.$ The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywhere. We prove that there is a gap between areas of non-smooth hyperbolic polyhedral surfaces and the area of smooth hyperbolic surfaces. The numerical result for the gap is obtained for hyperbolic polyhedral surfaces, homeomorphic to the double torus, whose 1-skeletons are cubic graphs.Comment: 23 pages, 3 figures. arXiv admin note: text overlap with arXiv:1804.1103

Cone Types and Spectral Radius of Hyperbolic Triangle Groups and Hyperbolic Tessellations

This paper is devoted to the study of tessellations of the hyperbolic plane, especially the ones associated to hyperbolic triangle groups $\Delta(l,m,n)$. We give a full description of the cone types of these graphs and show that their number depends only on the defining parameters of the group. We then use the cone types structure to provide estimates of the spectral radius for the simple random walk on these tessellations, from above and from below.Comment: 15 pages, 4 figure