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We show that it is possible to extend, in a homomorphic fashion, each quasisym- metric homeomorphism of the real line to a quasi-isometry of the upper-half plane. Epstein and Markovic have recently shown that a homomorphic extension to quasiconformal homeomorphisms of the upper-half plane is not possible.

Quasi-isometric extensions of quasisymmetric mappings of the real line compatible with composition