Hecke surfaces and duality transformations in lattice spin systems

We discuss two related subjects: (1) Hecke surfaces and $K$-regular graphs and (2) duality transformations for generalized Potts models. Each is related to deep mathematical and physical theories; at a first glance, they have nothing in common. But it has recently become more evident that there are deep internal relations between these two problems. The role of Hecke groups is especially interesting and mysterious in this context. We consider a few examples. The exposition is basically descriptive.