The Length Spectrum of a Flat Metric
Topics in Teichmuller Theory and Kleinian Groups
November 13, 2007 04:00 PM to 05:00 PM
Speakers:
Duchin, Moon
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Abstract: |
For hyperbolic surfaces, knowing how long all curves are
(and which curves correspond to which lengths) is enough to determine
the Poincare metric. In fact, we know from work of Hamenstadt that
6g-5 curves suffice on the surface of genus g>1. In Outer space, it
is also true that knowing lengths for all (conjugacy classes of) words
in the free group is enough to determine a marked graph. There,
however, Smillie and Vogtmann showed that no finite collection
suffices: for any finite set of words, there is a large family of
marked graphs for which the lengths of those words is the same on each
graph. It is natural to ask the same for singular flat structures on
hyperbolic surfaces, which arise in the study of Teichmuller geometry.
We show that the (marked) length spectrum does determine the flat
metric, but no finite set of curves suffices. In fact, to be
"spectrally rigid," a set of curves must be dense in PMF. This is
joint work with Chris Leininger
and Kasra Rafi. |
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