Short curves in hyperbolic 3-manifolds via knots on Heegaard surfaces

Ora Inizio: 
Ora Fine: 
Alessandro Sisto
ETH Zurigo

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.