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Session 37. Wild algebraic and geometric topology
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One-dimensional geodesic spaces, Part II: Marked length rigidity |
David Constantine, Wesleyan University, USA
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We prove a marked-length spectrum rigidity result for geodesic length spaces of topological dimension one. We show that for two such spaces, a homomorphism \(\phi: \pi_1(X)\to \pi_1(Y)\) with the property that the lengths of the geodesic representatives of \(\gamma\) and \(\phi(\gamma)\) are the same implies an isometry between the \(\pi_1\)-hulls of \(X\) and \(Y\) which induces the underlying homomorphism \(\phi\). This form of spectral rigidity holds in several `non-wild' geometric settings, such as for non-positively curved compact surfaces and compact locally symmetric spaces. This talk builds on the structure theory for the \(\pi_1\)-hulls developed in the talk ``One-dimensional geodesic spaces, Part I: Structure theory'' by Jean-François Lafont, also in this session.
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