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.
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Anderson, J.W., Bonfert-Taylor, P. & Taylor, E.C. Convergence Groups, Hausdorff Dimension, and a Theorem of Sullivan and Tukia. Geometriae Dedicata 103, 51–67 (2004). https://doi.org/10.1023/B:GEOM.0000013844.35478.e5
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DOI: https://doi.org/10.1023/B:GEOM.0000013844.35478.e5