On the IRS compactification of moduli space

TL;DR

This paper explores the relationship between the IRS compactification of the moduli space of hyperbolic surfaces and the classical Deligne-Mumford compactification, establishing a continuous finite-to-one correspondence with bounded fiber cardinalities.

Contribution

It constructs a continuous finite-to-one surjection from the augmented moduli space to the IRS compactification, linking two different compactification methods.

Findings

Established a continuous finite-to-one surjection

Bounded the fiber cardinalities uniformly

Connected IRS compactification with classical moduli space

Abstract

In arXiv:1503.08402v2 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.