Effective length‐projection bounds for hyperbolic 3‐manifolds diffeomorphic to S×R

We give a formula with explicit constants relating the subsurface projection of the end invariants of a hyperbolic 3‐manifold diffeomorphic to and the length of the geodesic representative in of the multicurve . This makes effective and computable the large projections versus short curves relation proved by Minsky. We give an application to closed hyperbolic 3‐manifolds fibering over the circle providing a geometric analog of the uniform projection bound in fibered faces of Minsky and Taylor.