Non-rectifiable limit sets of dimension one

Abstract

We construct 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. It is known that no divergence type group has such a deformation. The main tools in this construction are (1) a characterization of tangent points in terms of Peter Jones' $\beta$'s, (2) a result of Stephen Semmes that gives a Carleson type condition on a Beltrami coefficient which implies rectifiability and (3) a construction of quasiconformal deformations of a surface which shrink a given geodesic and whose dilatations satisfy an exponential decay estimate away from the geodesic.