Isospectral surfaces with distinct covering spectra via Cayley Graphs

Details Isospectral surfaces with distinct covering spectra via Cayley Graphs Isospectral surfaces with distinct covering spectra via Cayley Graphs

2010-06-28

arxiv.org/abs/1006.5414v1

Mathematics

Differential Geometry

12 pages, 5 figures

Scientific paper

The covering spectrum is a geometric invariant of a Riemannian manifold, more generally of a metric space, that measures the size of its one-dimensional holes by isolating a portion of the length spectrum. In a previous paper we demonstrated that the covering spectrum is not a spectral invariant of a manifold in dimensions three and higher. In this article we give an example of two isospectral Cayley graphs that admit length space structures with distinct covering spectra. From this we deduce the existence of infinitely many pairs of Sunada-isospectral surfaces with unequal covering spectra.

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