In 2004, Sormani and Wei introduced the covering spectrum: a
geometric invariant that isolates part of the length spectrum of a
Riemannian manifold. In their paper they observed that certain
Sunada isospectral manifolds share the same covering spectrum,
thus raising the question of whether the covering spectrum is a
spectral invariant. In the present paper we describe a group theoretic
condition under which Sunada’s method gives manifolds with
identical covering spectra. When the group theoretic condition of
our method is not met, we are able to construct Sunada isospectral
manifolds with distinct covering spectra in dimension 3 and
higher. Hence, the covering spectrum is not a spectral invariant.
The main geometric ingredient of the proof has an interpretation
as the minimum-marked-length-spectrum analogue of Colin
de Verdière’s classical result on constructing metrics where the
first $k$ eigenvalues of the Laplace spectrum have been prescribed.