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

Convergence groups, Hausdorff dimension, and a Theorem of Sullivan and Tukia
Convergence groups, Hausdorff dimension, and a Theorem of Sullivan and Tukia
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.
convergence groups, quasiconformal mappings, hausdorff dimension
51-67
Anderson, J.W.
739c0e33-ef61-4502-a675-575d08ee1a98
Bonfert-Taylor, P.
f8fada49-f840-4e4a-ab7c-12fc69d5818f
Taylor, E.C.
c6267a58-9224-42a9-a938-00068c526657
Anderson, J.W.
739c0e33-ef61-4502-a675-575d08ee1a98
Bonfert-Taylor, P.
f8fada49-f840-4e4a-ab7c-12fc69d5818f
Taylor, E.C.
c6267a58-9224-42a9-a938-00068c526657

Anderson, J.W., Bonfert-Taylor, P. and Taylor, E.C. (2004) Convergence groups, Hausdorff dimension, and a Theorem of Sullivan and Tukia. Geometriae Dedicata, 103 (1), 51-67. (doi:10.1023/B:GEOM.0000013844.35478.e5).

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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.

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Published date: 2004
Keywords: convergence groups, quasiconformal mappings, hausdorff dimension

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Local EPrints ID: 29878
URI: https://eprints.soton.ac.uk/id/eprint/29878
PURE UUID: 7c209464-448f-4b4f-99ef-805515502dce
ORCID for J.W. Anderson: ORCID iD orcid.org/0000-0002-7849-144X

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Date deposited: 12 May 2006
Last modified: 15 Aug 2019 00:53

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Author: J.W. Anderson ORCID iD
Author: P. Bonfert-Taylor
Author: E.C. Taylor

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