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I will speak about work with Guofang Wei of UC Santa Barbara.
In 2004, we defined a new concept called
the covering spectrum which roughly measures the size of one
dimensional "holes" in a Riemannian manifold and demonstrated
that, on a compact space, the covering spectrum is a subset of
the (1/2) length spectrum, is determined by the marked length
spectrum and can be used to select a special sequence of
generators of the fundamental group. The covering spectrum was
defined using a sequence of covering spaces, called the
delta covers, and we proved that the universal cover is
such a delta cover.
None of these results hold for complete noncompact spaces even
when one adds the condition that the spaces are locally compact.
Recently, we have proven corresponding theorems for complete
spaces, providing examples to show they are exact. In
addition we have defined and analyzed two new spectra: the cut-off
covering spectrum which measures holes that do not have
the loops to infinity property, and the rescaled covering
spectrum which measures holes that grow linearly at
infinity.
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