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.

Anahtar Kelimeler:
## Mapping class group, growth series, growth functions

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.

Keywords:
## Mapping class group, growth series, growth functions,

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- Ferihe ATALAN Received 27.01.2009
- Department of Mathematics, Atilim University, 06836 Ankara-TURKEY
- e—mail: fatalan©atilim.edu.tr Mustafa KORKMAZ
- Department of Mathematics,
- Middle East Technical University, 06531 Ankara-TURKEY
- e—mail: korkmaz©metu.edu.tr

**ISSN:**1300-0098**Yayın Aralığı:**6**Yayıncı:**TÜBİTAK

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