We prove that in the pure mapping class group of the 3-punctured projective plane equipped with the word metric induced by certain generating set, the ratio of the number of pseudo-Anosov elements to the number of all elements in a ball centered at the identity tends to one, as the radius of the ball tends to infinity. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 524 – 534 ¨ ITAK ˙ c TUB ⃝ doi: Counting pseudo-Anosov mapping classes on the 3-punctured projective plane Bla˙zej SZEPIETOWSKI∗ Institute of Mathematics, Gda´ nsk University, Wita Stwosza 57, 80-952 Gda´ nsk, Poland Received: • Accepted: • Published Online: • Printed: Abstract: We prove that in the pure mapping class group of the 3-punctured projective plane equipped with the word metric induced by certain generating set, the ratio of the number of pseudo-Anosov elements to the number of all elements in a ball centered at the identity tends to one, as the radius of the ball tends to infinity. We also compute growth functions of the sets of reducible and pseudo-Anosov elements. Key words: Mapping class group, nonorientable surface, growth functions 1. Introduction Let G be a group with a finite generating set A . For x ∈ G the length of x with respect to A is defined to be the minimum number of factors needed to express x as a product of elements of A and their inverses. We denote it by ||x||A . The word metric on G with respect to A is defined as dA (x, y) = ||xy −1 ||A for x, y ∈ G . For a subset X ⊂ G , the growth function of X with respect to A is the function f (z) defined by the power series ∞ ∑ Cn z n , where the coefficient Cn is equal to the number of elements of length n in X . The density n=0 d(X) of X with respect to A is defined as d(X) = lim n→∞ #(B(n) ∩ X) , #B(n) where B(n) is the set of elements of G of length at most n (it is the ball of radius n, centered at the identity, with respect to the word metric induced by A ), and # denotes the cardinality. Let S be a compact surface with a finite set P of distinguished points in the interior of S called punctures. We denote as Homeo(S, P ) the topological group of all, orientation preserving if S is .
