Young person's guide to translation surfaces of genus two: McMullen's connected sum theorem (Q1951339)

In this expository note, the authors give another proof of a theorem of \textit{C. T. McMullen} [Ann. Math. (2) 165, No. 2, 397--456 (2007; Zbl 1131.14027)] on translation surfaces in genus 2: Every translation surface of genus 2 can be represented as a connected sum of two tori glued along a geodesic segment. Their approach consists in an elementary case-by-case distinction combined with the fact that every such surface has a representation via a so-called polyband construction. Alongside they reprove that every translation surface of genus 2 can be represented as a centrally symmetric polygon with side-identifications by translations, and thus admits an involution.