Non-rectifiable limit sets of dimension one. (Q1394488)

The author considers and constructs quasiconformal deformations of convergence type Fuchsian groups such that the resulting limit set is a Jordan curve of Hausdorff dimension 1, but having tangents almost nowhere. He answers the question, asked by \textit{M. Zinsmeisters} and \textit{K. Astala} [C. R. Acad. Sci., Paris, Sér. I 311, 301--306 (1990; Zbl 0709.30039)], that whether there are quasi-Fuchsian groups whose limit set are not locally rectifiable, but still have dimension 1, by showing that there are many such groups.