# Counting essential surfaces in 3-manifolds

@article{Dunfield2020CountingES, title={Counting essential surfaces in 3-manifolds}, author={Nathan M. Dunfield and Stavros Garoufalidis and J. Hyam Rubinstein}, journal={Inventiones mathematicae}, year={2020}, volume={228}, pages={717 - 775} }

We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected) essential surfaces in terms of their Euler characteristic always has a short generating function and hence has quasi-polynomial behavior. This gives remarkably concise formulae for the number of such surfaces, as well as detailed asymptotics. We give…

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