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Graeme O'Brien

Doctor of Philosophy, (Mathematics)
Study Completed: 2018
College of Sciences


Thesis Title
Random discrete groups of mobius transformations

Read article at Massey Research Online: MRO icon

Hyperbolic spaces have constant negative curvature and the study of discrete groups of transformations on these spaces allows their geometry to be investigated via quotient manifolds. These could be spheres, tori or other much stranger structures. Decomposition via discrete groups is not only important to mathematicians but to understanding the physical structure of the universe itself. Mr O’Brien’s research offers a natural probability distribution on random Möbius transformations. Calculations can be made for the first time that put actual numbers to what proportion of groups are discrete, what proportion of each quotient topology one might have and what the dimension of the limit sets might be. Although the concepts have been applied to two dimensional surfaces the approach suggests a way forward for three dimensions as well.

Distinguished Professor Gaven Martin
Associate Professor Shaun Cooper