Isometries of real hyperbolic spaces correspond to Mobius transformations of their spheres at infinity.  Thanks to Paulin and Bonk-Schramm, quasi-isometries of Gromov hyperbolic spaces correspond to quasisymmetries between their boundaries at infinity.  After discussing this background, I'll describe how certain maps between relative hyperbolic groups correspond to maps between their Bowditch boundaries, and an application.  Joint work with Alessandro Sisto.

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