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We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary of the solvable Lie groups) are biLipschitz with respect to the visual metric. Mathematics Subject Classification 2010). 20F65, 30C65, 53C20.doi:10.5169/seals-323262 fatcat:f2uzgj2iivffbo3sipzdmdrd7i