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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.

MSC:

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