The curious moduli spaces of unmarked Kleinian surface groups

TL;DR

This paper introduces a new moduli space of unmarked hyperbolic 3-manifolds related to a fixed surface, revealing complex local topology and connecting it with classical surface moduli spaces through embeddings and compactifications.

Contribution

It defines a novel 3-dimensional moduli space AI(S) for hyperbolic 3-manifolds and explores its topological properties and relationships with the classical moduli space M(S).

Findings

AI(S) has bizarre local topology with many non-closed points.

There exists a natural embedding of M(S) into AI(S).

A compactification of AI(S) extends the embedding of M(S).

Abstract

Fixing a closed hyperbolic surface S, we define a moduli space AI(S) of unmarked hyperbolic 3-manifolds homotopy equivalent to S. This 3-dimensional analogue of the moduli space M(S) of unmarked hyperbolic surfaces homeomorphic to S has bizarre local topology, possessing many points that are not closed. There is, however, a natural embedding of M(S) into AI(S) and a compactification of AI(S) such that this embedding extends to an embedding of the Deligne-Mumford compactification of M(S) into the compactification of AI(S).