## 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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