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It is known that the Burau representation of the 4-strand braid group isfaithful if and only if certain matrices f and k generate a (non-abelian) freegroup. Regarding f and k as isometries of a euclidean building we show that f^3and k^3 generate a free group. We give two proofs, one utilizing the metricgeometry of the building, and the other using simplicial retractions.

A free subgroup in the image of the 4-strand Burau representation