Hecke graphs, Ramanujan graphs and generalized duality transformations for lattice spin systems

I discuss two related subjects, 1) Hecke surfaces and K-regular graphs, and 2) duality transformations for general Potts models. Each of them is related to deep mathematical and physical theories, and at a first glance, they have nothing in common. However, it became evident in recent years that there exist deep internal relations between these two problems. Especially interesting (and mysterious) is the role of Hecke groups in this context. I consider a few examples of relevant topics: A. Hecke surfaces with large cusps. B. Hecke graphs and Ramanujan graphs.