Counting incompressible surfaces in 3-manifolds

Abstract

Counting embedded curves on a hyperbolic surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting incompressible surfaces in a hyperbolic 3-manifold, with the key difference that now the surfaces themselves have more intrinsic topology. As there are only finitely many incompressible surfaces of bounded Euler characteristic up to isotopy in a hyperbolic 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Haken’s normal surface theory and facts about branched surfaces, we can characterize not just the rate of growth but show it is (essentially) a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples. This is joint work with Stavros Garoufalidis and Hyam Rubinstein. Based on https://arxiv.org/abs/2007.10053

Speaker

Nathan Dunfield has been at University of Illinois at Urbana-Champaign since 2007. Previously, He spent four years at Harvard and four years at Caltech after getting my PhD from the University of Chicago sometime back in the 20th century. In 2013, He became a Fellow of the American Mathematical Society. His research area is the topology and geometry of 3-manifolds.

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https://nmd.web.illinois.edu/