Virtually Haken conjecture

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.