Counting incompressible surfaces in three-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 three-manifold, with the key difference that now the surfaces themselves have a more intrinsic topology. As there are only finitely many incompressible surfaces of bounded Euler characteristic up to isotopy in a hyperbolic three-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.

differential geometrydynamical systemsgroup theorygeometric topology

Audience: researchers in the topic

( slides | video )

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Geometry and topology online

Series comments: You can also find up-to-date information on the seminar homepage - warwick.ac.uk/fac/sci/maths/research/events/seminars/areas/geomtop/

The talks start five minutes after the hour. Talks are typically 55 minutes long, including time for questions.

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