An invariant random subgroup is a conjugation-invariant probability measure on the space of closed subgroups of a given group. Such measures may be viewed as a common generalization of normal subgroups and lattices.

Every finite-area hyperbolic surface determines an invariant random subgroup of the group of orientation-preserving isometries of the hyperbolic plane. In this way, the moduli space of the surface embeds into the space of invariant random subgroups. Taking the closure of its image produces a compactification known as the IRS compactification. Krifka proved that there is a continuous finite-to-one map from the augmented moduli space onto this compactification.

The main goal of the talk is to construct metrics on the moduli space of the once-punctured torus using the invariant-random-subgroup approach. We will then discuss their behaviour when a simple closed geodesic is pinched and compare the resulting asymptotic estimates with those for the classical Weil--Petersson metric.