We introduce a geometrically natural probability measure
$\mu $ on the
group PSL$\left(2,\mathbb{R}\right)$,
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
twogenerator 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$\left(2,\mathbb{R}\right)$
(that is, selected via
$\mu $), then
$0.85<$Pr{$\langle \overline{f,g}\rangle =PSL\left(2,\mathbb{R}\right)$}$<0.9$;
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$\left(2,\mathbb{R}\right)$
conditioned to be hyperbolic, then Pr$\left\{\langle \overline{f,g}\rangle =PSL\left(2,\mathbb{R}\right)\right\}=12\u2215{n}^{2}$
(Theorem 12.5). We establish results such as the p.d.f. for the translation length
${\tau}_{f}$ of a random
hyperbolic to be
$H\left[\tau \right]=4\u2215{\pi}^{2}\phantom{\rule{2.77626pt}{0ex}}tanh\frac{\tau}{2}\phantom{\rule{2.77626pt}{0ex}}logtanh\phantom{\rule{0.3em}{0ex}}\frac{\tau}{4}$
(Theorem 4.9), along with related geometric invariants.
