Abstract: There are various ways of defining the "width" of a 3-dimensional manifold. Well-known examples include the Heegaard genus, or, in case of hyperbolic 3-manifolds, the volume. Driven by the algorithmic study of 3-manifolds, recent years have seen a growing interest in combinatorial notions of width defined through triangulations: it has been shown that several computationally hard problems about 3-manifolds can be efficiently solved for triangulations that are sufficiently "thin" in a certain sense.
In this talk we give an overview of recent results that link these combinatorial width parameters with classical topological invariants of 3-manifolds in a quantitative way. To establish our theorems, we rely on generalized Heegaard splittings and on layered triangulations.
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