Teichmüller spaces, triangle groups and Grothendieck dessins

This survey article considers moduli of algebraic curves using techniques from the complex analytic Teichmüller theory of deformations for the underlying Riemann surfaces and combinatorial topology of surfaces. The aim is to provide a readable narrative, suitable for people with a little background in complex analysis, hyperbolic plane geometry and discrete groups, who wish to understand the interplay of combinatorial, geometric and topological processes in this area. We explore in some detail a natural relationship with Grothendieck dessins, which provides both an appropriate setting in which to describe Veech curves (a special type of Teichmüller disc) and also a framework for relating complex moduli to arithmetic data involving a field of definition for the associated algebraic curves.