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arXiv · 1712.03602

Random Kleinian Groups I: Random Fuchsian Groups

Gaven Martin·Graeme O'Brien
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Abstract

We introduce a geometrically natural probability measure on the group of all M\"obius transformations of the circle. Our aim is to study "random" groups of M\"obius transformations, and in particular random two-generator groups. By this we mean groups where the generators are selected randomly. The probability measure in effect establishes an isomorphism between random $n$-generators groups and collections of $n$ random pairs of arcs on the circle. Our aim is to estimate the likely-hood that such a random group is discrete, calculate the expectation of their associated parameters, geometry and topology, and to test the effectiveness of tests for discreteness such as J{\o}rgensen's inequality.

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Gaven Martin, Graeme O'Brien. 2017-12-10. Random Kleinian Groups I: Random Fuchsian Groups. https://arxiv.org/abs/1712.03602

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