Thurston’s compactification via geodesic currents: the case of non-compact finite area surfaces

In 1988, Bonahon gave a construction of Thurston's compactification of Teichmüller space using geodesic currents. His argument only applies in the case of closed surfaces, and there are good reasons for that. We present a variant which applies to surfaces of finite area and to do so we prove a control theorem for sequences of random geodesics. Note that this theorem may be of independant interest, especially when the surface is non-compact.