Vol. 303, No. 2, 2019

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

Gaven Martin and Graeme O’Brien

Vol. 303 (2019), No. 2, 669–701

We introduce a geometrically natural probability measure μ on the group PSL(2, ), 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 PSL(2, ) (that is, selected via μ), then 0.85 < Pr{f,g¯ = PSL(2, )} < 0.9; thus the probability the group is discrete is at least 1 10 (Theorem 8.3) and this increases to 2 5 if we condition the selection to hyperbolic elements (Theorem 11.6). Further, if ζ is a primitive n-th root of unity, n 2, and f(z) = ζz is the elliptic of order n, and we choose g PSL(2, ) conditioned to be hyperbolic, then Pr{f,g¯ = PSL(2, )} = 1 2n2 (Theorem 12.5). We establish results such as the p.d.f. for the translation length τf of a random hyperbolic to be H[τ] = 4π2 tanh τ 2 logtanhτ 4 (Theorem 4.9), along with related geometric invariants.

random discrete Möbius groups
Mathematical Subject Classification 2010
Primary: 22E40, 30F40
Received: 4 November 2018
Revised: 11 July 2019
Accepted: 11 July 2019
Published: 4 January 2020
Gaven Martin
Institute for Advanced Study
Massey University
New Zealand
Graeme O’Brien
Institute for Advanced Study
Massey University
New Zealand