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The Virtually Haken Conjecture is a conjecture in the field of geometric topology, specifically related to 3-manifolds. It posits that every closed, irreducible 3-manifold that has a fundamental group that is a free product of finitely many non-trivial groups is "virtually Haken." To unpack this, a few definitions are necessary: 1. **Closed 3-manifold**: A 3-manifold that is compact and without boundary.

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  1. 3-manifolds
  2. Geometric topology
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