2020/03/25 by Irmer, Ingrid
#FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2003.11212
A topology is defined on the mapping class group of a compact connected orientable surface. It is shown that a notion of "genericity" on subsets of the mapping class group arises from this definition. Many plausible results follow from this notion easily; for example, the set of pseudo-Anosov maps is shown to be generic, and can be assumed to have arbitrary large stretch factor, generically. Let M be a 3-manifold obtained from a Heegaard splitting of fixed genus g and generic gluing map. It is shown that for such manifolds, generically M is hyperbolic, has first Betti number zero and Heegaard genus exactly equal to g.