Alessandro Sisto
Given a Heegaard splitting of a manifold M, one can consider simple
closed curves on the Heegaard surface as elements of its curve graph. I
will discuss the result that if some such curve K is far from both disk
sets (as measured in the curve graph), then the complement M-K is
hyperbolic. Moreover, there is a condition involving subsurface
projection that further ensures that M is obtained by long Dehn filling
of M-K, yielding that M is hyperbolic and (the geodesic representative
of) K is short.
If the gluing map of the Heegaard splitting is chosen using a random
walk, then with high probability there exists a curve K satisfying the
required conditions. Hence, this proves a (Perelman-free)
hyperbolisation result for "generic" 3-manifolds, as well as providing
information about the injectivity radius.
Joint with Peter Feller.