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.
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Nageswari Shanmugalingam, Xiangdong Xie, A rigidity property of some negatively curved solvable Lie groups. Comment. Math. Helv. 87 (2012), no. 4, pp. 805–823DOI 10.4171/CMH/269