On the IRS compactification of moduli space

Abstract

In [Gel15] Gelander described a new compactification of the moduli space of finite area hyperbolic surfaces using invariant random subgroups. The goal of this paper is to relate this compactification to the classical augmented moduli space, also known as the Deligne-Mumford compactification. We define a continuous finite-to-one surjection from the augmented moduli space to the IRS compactification. The cardinalities of this map's fibers admit a uniform upper bound that depends only on the topology of the underlying surface.