2026-09-24T08:55:51Zhttps://mro.massey.ac.nz/server/oai/requestoai:mro.massey.ac.nz:10179/130082025-11-11T14:19:07Zcom_10179_294col_10179_296
DSpace at Massey University
author
O'Brien, Graeme K
2018-04-06T00:24:28Z
2018-04-06T00:24:28Z
2017
http://hdl.handle.net/10179/13008
Q112158900
https://www.wikidata.org/wiki/Q112158900
This thesis addresses two areas related to the quantification of discrete groups. We study "random" groups of Möbius transformations and in particular random two-generator groups; that is, groups where the generators are selected randomly. Our intention is to estimate the likelihood that such groups are discrete and to calculate the expectation of their associated geometric and topological parameters. Computational results of the author [55] that indicate a low probability of a random group being discrete are extended and we also assess the expected Hausdorff dimension of the limit set of a discrete group. In both areas of research analytic determinations are correlated with computational results. Our results depend on the precise notion of randomness and we introduce geometrically natural probability measures on the groups of all Möbius transformations of the circle and the Riemann sphere.
en
The Author
Möbius transformations
Discrete groups
Research Subject Categories::MATHEMATICS
Random discrete groups of Möbius transformations : probabilities and limit set dimensions : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
Thesis
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
URL
https://mro.massey.ac.nz/bitstreams/0374ef0c-6341-4845-94c0-06ab51de538d/download
File
MD5
179c59465df402be9502d618586dfe4d
1859622
application/pdf
01_front.pdf
URL
https://mro.massey.ac.nz/bitstreams/c6e1a5e2-6b15-49ea-ab9a-c324d136d64e/download
File
MD5
ed3e09241d454c6f342fa999811505c3
2644614
application/pdf
02_whole.pdf
URL
https://mro.massey.ac.nz/bitstreams/e47fe9a7-bc2f-4454-976a-688cd2ce3cfe/download
File
MD5
17284efc1e688c7e97c3f414bae7020f
9441
text/plain
01_front.pdf.txt
URL
https://mro.massey.ac.nz/bitstreams/50a3e523-2005-4c79-a170-4ae85df1f7c3/download
File
MD5
9f33cc68a255060cc7085d29ebd59c30
103523
text/plain
02_whole.pdf.txt