The spherical growth series of amalgamated free products of infinite cyclic groups

Abstract

Let $n$ be an integer greater than $1$. We consider a group presented as $G(p_1,p_2,\dots,p_n)=\langle x_1,x_2,\dots, x_n \mid x_1^{p_1} =x_2^{p_2}=\cdots =x_n^{p_n} \rangle$, with integers $p_1,p_2,\dots,p_n$ satisfying $2 \leq p_1 \leq p_2 \leq \cdots \leq p_n$. This group is an amalgamated free product of infinite cyclic groups and is geometrically realized as the fundamental group of a Seifert fiber space over the 2-dimensional disk with $n$ cone points whose associated cone angles are $\frac{2\pi}{p_1},\frac{2\pi}{p_2},\dots,\frac{2\pi}{p_n}$. In this paper, we present a formula for the spherical growth series of the group $G(p_1,\dots,p_n)$ with respect to the generating set $\{x_1,\dots,x_n,x_1^{-1},\dots,x_n^{-1}\}$. We show that from this formula, a rational function expression for the spherical growth series of $G(p_1,\dots,p_n)$ can be derived in concrete form for given $p_1,\dots,p_n$. In fact, we wrote an elementary computer program based on this formula that yields an explicit form of a single rational fraction expression for the spherical growth series of $G(p_1,\dots,p_n)$. We present such expressions for several tuples $(p_1,\dots,p_n)$. In 1999, C. P. Gill obtained a similar formula for the same group in the case $n=2$ and showed that there exists a rational function expression for the spherical growth series of $G(p_1,\dots,p_n)$ for $n \geq 2$.