Spectra related to the length spectrum

We extend the Covering Spectrum ($\mathrm{CS}$) of Sormani–Wei to two spectra, called the Extended Covering Spectrum ($\mathrm{ECS}$) and Entourage Spectrum ($\mathrm{ES}$) that are new metric invariants related to the Length Spectrum for Riemannian manifolds. These spectra measure the “size” of certain covering maps called entourage covers that generalize the $\delta$-covers of Sormani–Wei. For Riemannian manifolds $M$ of dimension at least $3$, we topologically characterize entourage covers as those covers corresponding to subgroups of $\pi_1 (M)$ that are the normal closures of finite subsets. We show that $\mathrm{CS} \subset \mathrm{ES}\subset \mathrm{MLS}$, where $\mathrm{MLS}$ is the set of lengths of closed curves that are shortest in their free homotopy classes. For Riemannian manifolds these inclusions can be strict. Finally, we give equivalent definitions for any metric on a Peano continuum, including resistance metrics on fractals, which have a Laplace Spectrum, opening new fronts in the old problem of the relationship between the Laplace Spectrum and the Length Spectrum.

Keywords

length spectrum, covering spectrum, Laplace spectrum, resistance metrics on fractals

2010 Mathematics Subject Classification

Primary 58J53. Secondary 53C23, 57M10.

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Received 10 April 2019

Accepted 29 December 2020

Published 25 April 2022