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The number of surfaces of fixed genus embedded in a 3-manifold

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Anastasia Tsveitkova (Rutgers)

Abstract: It was noticed before that presence of embedded essential surfaces in a 3-manifold can give information about that manifold. However to construct, classify or count such surfaces is a non-trivial task. If 3-manifold is complement of an alternating link with n crossings in a 3-sphere, we previously showed that the number of genus-g closed surfaces is bounded by a polynomial in n. This was the first polynomial bound. We then extended the result to spanning surfaces, and to 3-manifolds obtained as Dehn fillings of alternating links. This was joint work with Joel Hass and Abigail Thompson. In the talk, I will discuss its generalizations. One generalization is a joint work with Jessica Purcell. It concerns any cusped 3-manifold that is complement of a link alternating on some embedded surface in an arbitrary 3-manifold. Another one is joint work with Marc Lackenby. There, we prove that for any closed hyperbolic 3-manifold, there are polynomially many genus-g surfaces in terms of hyperbolic volume of the manifold.