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Convergence Groups, Hausdorff Dimension, and a Theorem of Sullivan and Tukia

Research paper by James W. Anderson, Petra Bonfert-Taylor, Edward C. Taylor

Indexed on: 01 Feb '04Published on: 01 Feb '04Published in: Geometriae Dedicata



Abstract

We show that a discrete, quasiconformal group preserving ℍn 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 ℍn. This generalizes a result due separately to Sullivan and Tukia, in which it is further assumed that the group act isometrically on ℍn, 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.