AI 中文总结
本文证明固定有限图的正规覆盖谱测度原子部分相同，给出原子质量显式公式，推广物理猜想，并建立连续部分对数Hölder正则性。
AI 中文摘要
我们证明，固定有限图$G$的介于万有覆盖与极大阿贝尔覆盖之间的所有正规覆盖，其谱测度的原子部分相同。我们给出了原子质量的显式双重公式，该公式推广并统一了关于万有覆盖树和极大阿贝尔覆盖的已知结果。它还证明了物理学文献中关于双曲格子的一个猜想的推广。此外，我们还建立了谱测度连续部分的对数Hölder正则性。我们的证明结合了冯·诺依曼代数、匹配理论和单调标记方法的技术。在附录中，我们证明了一个可能具有独立意义的逆广义Gallai-Edmonds结构定理。
英文摘要
We show that all normal covers of a fixed finite graph $G$ sitting between the universal and maximal abelian covers share the same atomic part of their spectral measures. We give an explicit double formula for the mass of atoms which generalises and unifies known results on universal covering trees and maximal abelian covers. It also proves an extension of a conjecture in physics literature on hyperbolic lattices. Moreover, we also establish the logarithmic Hölder regularity of the continuous part of the spectral measures. Our proof combines techniques from von Neumann algebras, matching theory and the monotone labelling method. In an appendix, we prove a converse generalised Gallai-Edmonds structural theorem which might be of independent interest.
