The twist-cofinite topology on the mapping class group of a surface

Abstract

A topology is defined on the mapping class group $\mathrm{Mod}(\Sigma )$ of a compact, connected, orientable surface $\Sigma$. It is shown that a notion of “genericity” on subsets of $\mathrm{Mod}(\Sigma )$ 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 arbitrarily 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 $b_1(M)=0$, $M$ is hyperbolic and $M$ has Heegaard genus exactly $g$.