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Marked length rigidity of rank one symmetric spaces and their product. (English) Zbl 0997.53034
The author proves that if two Zariski dense representations (not necessarily discrete, nor faithful, nor co-finite volume), from a group \(G\) into \(\text{Iso}(X)\) where \(X\) is a rank one symmetric space, have the proportional marked length spectrum, then they are conjugate (Theorem A). As a generalization of this result, in Section 8 the author shows that a Zariski dense representation into the isometry group of the product of rank one symmetric spaces is determined by the marked cross ratio.

53C24 Rigidity results
53C35 Differential geometry of symmetric spaces
53C10 \(G\)-structures
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