A new formula for the spherical growth series of an amalgamated free product of two infinite cyclic groups

Abstract

We consider a group presented as G(p,q) = <x, y|x^p = y^q>, with integers p and q satisfying 2 ≤ p ≤ q. The group is an amalgamated free product of two infinite cyclic groups and is geometrically realized as the fundamental group of a Seifert fiber space over the 2-dimensional disk with two cone points whose associated cone angles are and . We present a formula for the spherical growth series of the group G(p,q) with respect to the generating set {x, y, x^-1, y^-1}, from which a rational function expression for the spherical growth series of G(p,q) is derived concretely, once p and q are given. In fact, an elementary computer program constructed from the formula yields an explicit form of a single rational fraction expression for the spherical growth series of G(p,q). Such expressions for several pairs (p,q) appear in this paper. In 1999, C. P. Gill already provided a similar formula for the same group. The formula given here takes a different form from his formula, because the method we used here is independent of that introduced by him.