Thurston's broken windows only theorem revisited

The'broken windows only theorem' is the main theorem of the third paper among a series of the paper in which Thurston proved his uniformisation theorem for Haken manifolds. In this chapter, we show that the second statement of this theorem is not valid, giving a counter-example. 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 constitute a 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, although its weaker version, which is sufficient for the proof of the uniformisation theorem, had already been proved.