The bounded image theorem is a key step of Thurston’s uniformisation theorem for Haken manifolds, but no one knows how he proved it.
I will explain my guess on how he intended to prove it, whose argument is thought to be based on an incorrect statement of his « broken window only theorem » .
I will also present a sketch of a new proof involving modern Kleinian group theory, which is based on my joint work with Cyril Lecuire.