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Presented By: Topology Seminar - Department of Mathematics

Topology seminar: Marked length pattern rigidity and Busemann cocycle

Yanlong Hao

Given a closed Riemannian manifold M, the length of the shortest geodesic for each free homotopy class of loops on M is called the (minimal) length of the class. This gives a map called marked length spectrum. It is conjectured that the fundamental group and marked length spectrum together determine the isometric type of negatively curved manifolds. This conjecture has been verified for surfaces and locally symmetric spaces. In this talk, we show that for negatively curved arithmetic manifolds, the fundamental group with all pairs of different equal length classes, i.e., marked length pattern, is enough to recover the metric up to scaling. The central idea is marked length spectrum rigidity of cocycles.

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