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<p>We prove that every closed, connected, orientable surface <i>S</i> of negative Euler characteristic admits a pair of finite-degree covers which are length isospectral over <i>S</i> but generically not simple length isospectral over <i>S</i>. To do this, we first characterize when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations. We also construct hyperbolic surfaces <i>X</i> and <i>Y</i> with the same full unmarked length spectrum but so that for each <i>k</i>, the sets of lengths associated to curves with at most <i>k</i> self-intersections differ.</p>

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Unmarked simple length spectral rigidity for covers

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年2月24日

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  • Tarik Aougab,
  • Max Lahn,
  • Marissa Loving,
  • Nicholas Miller

摘要

We prove that every closed, connected, orientable surface S of negative Euler characteristic admits a pair of finite-degree covers which are length isospectral over S but generically not simple length isospectral over S. To do this, we first characterize when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations. We also construct hyperbolic surfaces X and Y with the same full unmarked length spectrum but so that for each k, the sets of lengths associated to curves with at most k self-intersections differ.