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Thurston’s Broken Windows Only Theorem Revisited

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In the Tradition of Thurston III

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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.

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Authors and Affiliations

  1. Gakushuin University, Mathematics Department, Tokyo, Japan

    Ken’ichi Ohshika

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  1. Ken’ichi Ohshika
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Correspondence to Ken’ichi Ohshika .

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Editors and Affiliations

  1. Department of Mathematics, Gakushuin University, Tokyo, Japan

    Ken’ichi Ohshika

  2. Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, Strasbourg, France

    Athanase Papadopoulos

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Ohshika, K. (2024). Thurston’s Broken Windows Only Theorem Revisited. In: Ohshika, K., Papadopoulos, A. (eds) In the Tradition of Thurston III. Springer, Cham. https://doi.org/10.1007/978-3-031-43502-7_8

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