Random Kleinian Groups I: Random Fuchsian Groups

TL;DR

This paper introduces a natural probability measure on Möbius transformations of the circle to study random Fuchsian groups, focusing on their geometric, topological properties, and discreteness likelihood.

Contribution

It develops a framework for analyzing random two-generator groups of Möbius transformations and evaluates their geometric and algebraic properties.

Findings

Provides a probability measure linking random groups to arcs on the circle

Estimates the likelihood of a random group being discrete

Calculates expected geometric and topological parameters

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.