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
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Date
2017
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Massey University
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Abstract
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.
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Möbius transformations, Discrete groups, Research Subject Categories::MATHEMATICS