Graduate Student Colloquium
Sam Dodds
UIC
Weird Flex: Rigidity and Flexibility of Groups
Abstract: Given an orientable surface S with genus at least 2, it is always possible to fix a metric on S which is locally isometric to the hyperbolic plane. Such a metric space is called a hyperbolic surface. Take a geodesic circle which separates S into two pieces, and spin one of the pieces. This results in a continuous family of hyperbolic surfaces that are homeomorphic but not isometric. One of the landmark geometric theorems of the 20th century was Mostow’s Rigidity Theorem: that for higher dimensions this is impossible; if two compact hyperbolic manifolds of dimension greater than 2 are homeomorphic, then they are isometric. It turns out that the flexibility and rigidity of these spaces really originates with their fundamental groups. The fundamental groups of surfaces are in some sense flexible groups, but in higher dimensions fundamental groups of hyperbolic manifolds are rigid. More precisely, they have flexible/rigid group actions. We will survey various rigidity theorems for some groups, outline some proofs (including a proof of Mostow’s theorem), and talk about flexibility results for other groups, (including some constructions).
biggest man. geometric algebra.
Monday March 14, 2022 at 5:00 PM in 636 SEO