Let $N$ be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold $M$. In this paper, we study the extent to which $N$ admits as much symmetry as $M$. Our main results are examples of $N$ that exhibit two extremes of behavior. On the one hand, we find $N$ with maximal symmetry, i.e. Isom($M$) acts on $N$ by isometries with respect to some negatively curved metric on $N$. For these examples, Isom($M$) can be made arbitrarily large. On the other hand, we find $N$ with little symmetry, i.e. no subgroup of Isom($M$) of "small" index acts by diffeomorphisms of $N$. The construction of these examples incorporates a variety of techniques including smoothing theory and the Belolipetsky-Lubotzky method for constructing hyperbolic manifolds with a prescribed isometry group.

中文翻译：

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奇异负弯曲流形的对称性

令$N$ 是一个与闭双曲流形$M$ 同胚但不微分同胚的光滑流形。在本文中，我们研究了 $N$ 承认与 $M$ 一样多的对称性的程度。我们的主要结果是表现出两种极端行为的 $N$ 示例。一方面，我们发现$N$ 具有最大对称性，即Isom($M$) 通过等距作用于$N$，相对于$N$ 上的一些负弯曲度量。对于这些示例，Isom($M$) 可以任意大。另一方面，我们发现$N$ 几乎没有对称性，即“小”索引的Isom($M$) 的子群没有$N$ 的微分同胚作用。这些示例的构造结合了多种技术，包括平滑理论和用于构造具有规定等距群的双曲流形的 Belolipetsky-Lubotzky 方法。