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 $\mathrm{\Delta }(l,m,n)$. 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.