Inspired by the central role geometric structures play in our understanding of the taxonomy of three-manifolds, we will explore the extent to which compact locally homogeneous three-manifolds are characterized up to universal Riemannian cover by their spectra. In this talk we will demonstrate that within the universe of compact locally homogeneous Riemannian manifolds, closed three-manifolds equipped with geometric structures modeled on six of the eight Thurston geometries are determined up to universal Riemannian cover by their spectra, a result that includes all compact locally symmetric three-manifolds and is optimal due to the existence of isospectral hyperbolic three-manifolds, for example. Furthermore, we show that any space modeled on the symmetric space $\mathbb{S}^{2} \times \mathbb{E}$ or Nil equipped with an arbitrary left-invariant metric is uniquely determined by its spectrum among all locally homogeneous spaces. These results follow from more general observations, regarding the eight “metrically maximal” three-dimensional geometries, that strongly suggest local geometry is “audible” among compact locally homogeneous three-manifolds.

This is joint work with Samuel Lin (Oklahoma) & Benjamin Schmidt (Michigan State)