Angle structures and representations through Dehn surgery space

Abstract: Let $T$ be an ideal triangulation of a three-manifold $M$. The shapes of the tetrahedra in the triangulation give gluing equations, the solutions to which we call the "shape variety". We look at the real locus of the shape variety where the tetrahedra are all flat. There are three types of angle structure that exist with flat tetrahedra.

After Thurston sketched the boundary of Dehn surgery space of the figure-eight knot, Hodgson broke it down into three types, which we denote as representations from the fundamental group of the knot complement into $\mathrm{PSL}(2,\mathbb{R})$, $\mathrm{PGL}(2,\mathbb{R})$, and $\mathrm{SO}(3)$.

What different angle structures have which types of representation? Can they then also be deformed into a positive volume manifold such that all tetrahedra are positive? We will look at all the different combinations and determine which ones can exist.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

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 at 13:30. Talks are typically fifty minutes long, with ten minutes for questions.