Genericity of pseudo-Anosov mapping classes, when seen as mapping classes

Abstract

We prove that pseudo-Anosov mapping classes are generic with respect to certain notions of genericity reflecting that we are dealing with mapping classes. More precisely, we consider a number of functions $\rho$ on the mapping class group, and show that the proportion of pseudo-Anosov mapping classes with $\rho$-value at most $R$ tends to 1 as $R$ tends to infinity. The functions we consider include measuring the complexity of mapping classes using quasi-conformal distortion or Lipschitz distortion. We present a uniform approach to this problem using geodesic currents.