Thurston's broken windows only theorem revisited (Q6610517)

Thurston planned to publish a proof of his celebrated uniformisation theorem for Haken 3-manifolds in a series of seven papers; only the first paper was officially published, and the second and third paper exist as preprints in his collected works. The ``broken windows only theorem'' is the main theorem of the third paper. ``In this chapter, we show that the second statement of this theorem is not valid, giving a counterexample. We also give a weaker version of this statement with a proof. In the last section, we speculate on how this second statement was intended to be used in the proof of the bounded image theorem which constituted a key of the uniformisation theorem. The proof of the bounded image theorem was obtained only quite recently (cf. [\textit{C. Lecuire} and \textit{K. Ohshika}, ``Thurston's Bounded image theorem'', Preprint, \url{arXiv:2209.05023}]) although a weaker version, which is sufficient for the proof of the uniformisation theorem, had already been proved.''\N\NFor the entire collection see [Zbl 1537.57002].