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Thurston’s Broken Windows Only Theorem Revisited

In: In the Tradition of Thurston III

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  • Ken’ichi Ohshika

    (Mathematics Department, Gakushuin University)

Abstract

The “broken windows only theorem” is the main theorem of the third paper among a series of papers 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 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, although a weaker version, which is sufficient for the proof of the uniformisation theorem, had already been proved.

Suggested Citation

  • Ken’ichi Ohshika, 2024. "Thurston’s Broken Windows Only Theorem Revisited," Springer Books, in: Ken’ichi Ohshika & Athanase Papadopoulos (ed.), In the Tradition of Thurston III, chapter 0, pages 275-297, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-43502-7_8
    DOI: 10.1007/978-3-031-43502-7_8
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