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Number of pseudo-Anosov elements in the mapping class group of a four-holed sphere

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. | Turk J Math 34 (2010) , 585 – 592. ¨ ITAK ˙ c TUB doi:10.3906/mat-0901-38 Number of pseudo–Anosov elements in the mapping class group of a four–holed sphere Ferihe Atalan and Mustafa Korkmaz Abstract 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. Key Words: Mapping class group, growth series, growth functions. 1. Introduction A finitely generated group can be seen as a metric space after fixing a finite generating set. The metric is the so called word metric. As is well-known, the mapping class group of a compact surface is finitely generated, thus a metric space. One of the purposes of this note is to prove that, after fixing a certain set of generators, in a ball centered at the identity in the pure mapping class group of a four holed sphere (which is a free group of rank two), almost all elements are pseudo–Anosov. More precisely, in a ball with center at the identity, the ratio of the number of pseudo–Anosov elements to the number of all elements tends to one as the radius of the ball tends to infinity. In fact, we prove more: We give the growth series of reducible and of pseudo–Anosov elements with respect to a fixed set of generators. It turns out that the growth functions of these elements are rational. This gives a partial answer to Question 3.13 and verifies Conjecture 3.15 in [2] in a special case. Similar results are proved in [5] and [6] by using different methods, which do not immediately imply the results of this paper. 2. Preliminaries Let G be a finitely generated group with a finite generating set A, so that every element of G can be written as a product of elements in A ∪ A−1 . The length of an element g ∈ G (with .