Surfaces and their mapping class groups are central objects in geometry and low dimensional topology. Their theory is closely related to the theory of Riemann surfaces, Teichmuller theory, topology and geometry of three and four dimensional smooth manifolds, symplectic manifolds and knot theory. Mapping class groups provide powerful tools in most of these theories. There are several ways of investigating algebraic and geometric properties of mapping class groups. The complexes of curves on surfaces proved very useful in the study of mapping class groups since their introduction by Harvey in...