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The curious moduli spaces of unmarked Kleinian surface group
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 1, February 2012
- pp. 71-85
- 10.1353/ajm.2012.0006
- Article
- Additional Information
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 $\overline{\cal M}}(S)
\rightarrow {\rm A}\overline{\cal I}}(S)$.