We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set. We prove that the ratio of the number of pseudo-Anosov elements to that of all elements in a ball with center at the identity tends to one as the radius of the ball tends to infinity.
ATALAN, FERİHE and KORKMAZ, MUSTAFA
"Number of pseudo--Anosov elements in the mapping class group of a four--holed sphere,"
Turkish Journal of Mathematics: Vol. 34:
4, Article 13.
Available at: https://journals.tubitak.gov.tr/math/vol34/iss4/13