Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

中文翻译：

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用 Haar 测度重新审视 Leighton 定理

Leighton 的图覆盖定理指出，一对具有同构全覆盖的有限图有一个共同的有限覆盖。我们提供了一个允许泛化的 Leighton 定理的新证明；我们证明了带有鳍的图的相应结果。作为推论，我们在 Cashen-Macura 和 Hagen-Touikan 的工作的基础上获得了具有线条图案的自由组的图案刚性。为了说明未来应用的潜力，我们给出了一个自由基团循环双倍家族的准等距刚性结果。