Decompositions of Riemann surfaces into cylinders

Classically there were two kinds of decomposition results
(1) The regular Jenkins-Strebel differentials exist on any Riemann surface of genus $\ge1$ and metricall present this surface as a union of finite straight cylinders.
(2) The Jenkins-Strebel differentials with prescribed poles of order 2 exist on any punctured Riemann surface that turns metrically into a union of semi-finite straight cylinders.
No connection of this construction to the construction (1) to the moduli of closed Riemann surfaces is known currently. As for the construction (2), the results of Kontsevich-Penner-Witten-... relate it to the canonical decomposition of somewhat extended moduli spaces of punctured Riemann surfaces, parametrized by embedded graphs (dessins d'enfants). Some striking – Fields prizes worth – applications of these decompositions are known. These and others will be discussed.