It has been believed that pseudo-Anosovs are generic in the mapping class group in certain aspects. One way to pick a random isometry is to consider a random walk on the mapping class group. In this setting, Maher proved that the probability for non-pseudo-Anosovs at step n decreases to 0. Later, Baik, Kim and I built upon Maher-Tiozzo's theory and proved that almost every sample path becomes pseudo-Anosov eventually. Meanwhile, one can instead pick a random isometry inside a radius R ball of the mapping class group and see the asymptotic proportion. This counting problem and the random walk estimates are widely different since one cannot count group elements exactly once from the latter perspective.
In this talk, I will explain how to establish the exponential genericity of pseudo-Anosovs in the counting problem via the random walk method. The essential ingredient is that if we generate the random walk with an 'almost Schottky' generating set, the probability for non-pseudo-Anosovs at step n decays exponentially. If time permits, I will explain how this strategy can be implemented to random walks on automatic structures of groups, following Gekhtman-Taylor-Tiozzo.