Convergence groups, Hausdorff dimension, and a Theorem of Sullivan and Tukia

Abstract

We show that a discrete, quasiconformal group preserving Hopf n has the property that its exponent of convergence and the Hausdorff dimension of its limit set detect the existence of a non-empty regular set on the sphere at infinity to Hopf n. This generalizes a result due separately to Sullivan and Tukia, in which it is further assumed that the group act isometrically on Hopf n, i.e. is a Kleinian group. From this generalization we are able to extract geometric information about infinite-index subgroups within certain of these groups