Building surfaces from equilateral triangles

This lecture by the instructor covers the construction of Riemann surfaces from equilateral triangles, exploring topics such as finite-type maps, equilateral triangulations, compact Belyi surfaces, and non-compact surfaces. The lecture delves into the dynamics of finite-type maps, the existence of finite-type maps on non-compact Riemann surfaces, Belyi functions, and the relationship between equilateral triangulations and Belyi functions. It also discusses the compact and non-compact cases, the existence of finite-type maps on non-compact Riemann surfaces, and various constructions involving quasiconformal folding and quasiregular functions.