The curious moduli spaces of unmarked Kleinian surface group

Abstract

Fixing a closed hyperbolic surface $S$, we define a moduli space ${\rm A}{\cal I}(S)$ of unmarked hyperbolic $3$-manifolds homotopy equivalent to $S$. This $3$-dimensional analogue of the moduli space ${\cal M}(S)$ of unmarked hyperbolic surfaces homeomorphic to $S$ has bizarre local topology, possessing many points that are not closed. There is, however, a natural embedding $\iota: {\cal M}(S) \rightarrow {\rm A}{\cal I}(S)$ and compactification ${\rm A}\overline{\cal I}}(S)$ such that $\iota$ extends to an embedding of the Deligne-Mumford compactification $\overline{\cal M}}(S) \rightarrow {\rm A}\overline{\cal I}}(S)$.