A long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show that any closed 4-manifold that carries a Thurston geometry and is not finitely covered by a product of two aspherical surfaces does not support (transitive) Anosov diffeomorphisms.

中文翻译：

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Thurston 几何四流形上的 Anosov 微分同胚

一个长期存在的猜想断言，封闭流形的任何 Anosov 微分同胚都被微分同胚有限覆盖，该微分同胚与幂零流形的双曲自同构拓扑共轭。在本文中，我们展示了任何带有 Thurston 几何形状且未被两个非球面的乘积有限覆盖的封闭 4-流形不支持（传递）Anosov 微分同胚。