Suppose that \((M,g)\) is a closed, negatively curved Riemannian manifold. Given another such manifold \((M,g')\) how do we determine whether \((M,g)\) and \((M,g')\) are the same (isometric)? It is conjectured that the isometry class of \((M,g)\) is determined by its marked length spectrum: the lengths of closed geodesics on \((M,g)\) ordered in a natural way. In this talk we discuss the following question. Is there a (hopefully ‘small’) set of closed geodesics that determines the full marked length spectrum of \((M,g)\)? That is, is there a small set of closed geodesics such that if \((M,g)\) and \((M',g')\) assign the same lengths to these geodesics then they must assign the same lengths to all closed geodesics (i.e. have the same marked length spectrum).
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Passcode: A group is called an ________ group if it admits an invariant mean. (8 letters, lowercase)