Random Möbius groups, I: Random subgroups of PSL$(2,\mathbb{R})$

Gaven Martin and Graeme O’Brien

Vol. 303 (2019), No. 2, 669–701

DOI: 10.2140/pjm.2019.303.669

Abstract

We introduce a geometrically natural probability measure $\mu$ on the group PSL$\left(2,ℝ\right)$, identified as the group of all Möbius transformations of the hyperbolic plane, which is mutually absolutely continuous with respect to the Haar measure. Our aim is to study topological generation and random subgroups, in particular random two-generator subgroups where the generators are selected randomly. This probability measure in effect establishes an isomorphism between random $n$-generator groups and collections of $n$ random pairs of arcs on the circle. Our aim is to estimate the likelihood that such a random group topologically generates (or, conversely, is discrete). We also want to calculate the precise expectation of associated parameters, the geometry and topology, and to establish the effectiveness of tests for discreteness. We achieve an interesting mix of bounds and precise results. For instance, if $f,g{\in }_{\ast }$PSL$\left(2,ℝ\right)$ (that is, selected via $\mu$), then $0.85<$Pr{$\langle \overline{f,g}\rangle =PSL\left(2,ℝ\right)$}$<0.9$; thus the probability the group is discrete is at least $\frac{1}{10}$ (Theorem 8.3) and this increases to $\frac{2}{5}$ if we condition the selection to hyperbolic elements (Theorem 11.6). Further, if $\zeta$ is a primitive $n$-th root of unity, $n\ge 2$, and $f\left(z\right)=\zeta z$ is the elliptic of order $n$, and we choose $g{\in }_{\ast }$PSL$\left(2,ℝ\right)$ conditioned to be hyperbolic, then Pr$\left\{\langle \overline{f,g}\rangle =PSL\left(2,ℝ\right)\right\}=1-2∕n^2$ (Theorem 12.5). We establish results such as the p.d.f. for the translation length ${\tau }_f$ of a random hyperbolic to be $H\left[\tau \right]=-4∕{\pi }^{2}tanh\frac{\tau }{2}logtanh\frac{\tau }{4}$ (Theorem 4.9), along with related geometric invariants.

Keywords

random discrete Möbius groups

Mathematical Subject Classification 2010

Primary: 22E40, 30F40

Milestones

Received: 4 November 2018

Revised: 11 July 2019

Accepted: 11 July 2019

Published: 4 January 2020