How helpful is hyperbolic geometry?

Abstract: Hyperbolic geometry serves dual roles at the intersection of group theory and three-manifold topology. It plays the hero of group theory — rescuing the field from a morass of uncomputability — but the anti-hero of low-dimensional topology—seemingly responsible for much of the complexity of three-manifolds. Where do these roles overlap?

I’ll give examples of group-theoretic invariants of three-manifolds (or knots) that are NP-hard to compute, even for three-manifolds (or knots) that are promised to be hyperbolic. The basic idea is to show that the right-angled Artin semigroups of reversible circuits (a kind of combinatorial abstraction of particularly simple computer programs) can be quasi-isometrically embedded inside mapping class groups. Recent uniformity results concerning the coarse geometry of curve complexes play a key role.

This is joint work with Chris Leininger that builds on previous work with Greg Kuperberg.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

( chat | 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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