Random methods in 3-manifold theory
  We show that for any integers k and g, with g at least two, there are infinitely many closed hyperbolic 3-manifolds which are integral homology spheres with Casson invariant k, and Heegaard genus equal to g. This existence result is shown using random methods, using a model of random 3-manifolds arising from random walks on the mapping class group of a closed orientable surface. 
Alexander Lubotzky, Joseph Maher and Conan Wu
math.GT math.GR math.PR