Cusp shape and fractional Dehn twists of fibred hyperbolic 3-manifolds

Given a fibred hyperbolic 3-manifold with boundary, we coarsely relate the Euclidean geometry of its cusps to the classical fractional Dehn twist coefficient of its monodromy. This result fits into the broader programme of coarsely describing the geometry of a hyperbolic 3-manifold via combinatorial data. We are thus able to study the hyperbolic geometry of certain fibred 3-manifolds under Dehn filling. For example, we find coarse volume estimates for sufficiently twisted braid closures in terms of their braid words. We also prove that for any open book decomposition of a fixed manifold (that is not a lens space or solid torus) with fibre of fixed Euler characteristic the fractional Dehn twist coefficient in some boundary component is uniformly bounded. Finally, we obtain applications to contact topology. We give a geometric criterion on the binding of an open book decomposition for the corresponding contact structure in a hyperbolic 3-manifold to be tight.