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Extending homeomorphisms of the real line to the upper half plane.

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Resumen: Consider the following question: is it possible to extend a 
quasisymmetric homeomorphism of the real line to a quasiconformal 
homeomorphism of upper-half plane such that the extension of a 
composition it is the composition of the extensions?

Answer to this question is already known in this precise situation, 
but there are cases where these problems still need an answer 
(positive or negative). In this talk I would like to present the 
result of recent collaborations that provides some advance in the 
specific case of BiLipschitz homeomorphisms.

Joint work with José Afonso Barrionuevo, Felipe Gonçalves and Victor Medeiros.