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.
Mapping class group, growth series, growth functions
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