Random Kleinian Groups I: Random Fuchsian Groups
arXiv:1712.03602
Abstract
We introduce a geometrically natural probability measure on the group of all Möbius transformations of the circle. Our aim is to study "random" groups of Möbius 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 -generators groups and collections of 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 Jørgensen's inequality.