Abstract

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 ι: M(S) → AI(S) and compactification AI(S) such that ι extends to an embedding of the Deligne-Mumford compactification M(S) → AI(S). For a closed oriented hyperbolic surface S, the moduli space M(S) of hyperbolic (or Riemann) surfaces homeomorphic to S is a familiar and interesting object. Hyperbolic 3-manifolds homotopy equivalent to S have also been intensely studied since the foundational work of Ahlfors and Bers. An analogous moduli space of (unmarked) hyperbolic 3-manifolds homotopy equivalent to S is rarely considered. Here we define such a topological moduli space AI(S), study a bit of its local topology, and define a natural

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