Bers’ Constant and the Hairy Torus

Abstract

Every compact Riemann surface can be decomposed into Y-pieces. What can we say about the lengths of the geodesics involved in such a decomposition? Bers [3,4] proved that there exists a decomposition with lengths less than some constant which depends only on the genus. Bers’ theorem has numerous consequences for the geometry of compact Riemann surfaces (see for instance Abikoff [1], Bers [4], Seppälä [1]). In this book we shall give the following applications of Bers’ theorem. In Chapter 6 it gives a rough fundamental domain for the Teichmüller modular group, in Chapter 10 it is used in the proof of Wolpert’s theorem, and in Chapter 13 we apply Bers’ theorem to estimate the number of pairwise non-isometric isospectral Riemann surfaces possible.