The spherical growth series of amalgamated free products of infinite cyclic groups
Michihiko Fujii, Takuya Sakasai
TL;DR
The paper develops a general framework to compute the spherical growth series of G(p_1,...,p_n), an amalgamated free product of infinite cyclic groups realized as the fundamental group of a Seifert fiber space. Central to the approach is the Garside structure on a positive monoid and the modified normal form, from which a suitable-spread procedure yields geodesic representatives. The authors partition geodesics into canonical types and construct a bijective set of representatives Gamma, enabling explicit rational expressions for the spherical growth series in terms of the functions f, g, h and T_u. They provide a unified formula for all n ≥ 2, include parity-based case analyses (even/odd p_k - p_1), and supply concrete rational expressions for numerous explicit tuples, together with a Mathematica program for automated computation. This advances understanding of growth in Seifert-type amalgams and yields practical computations of growth series via rational functions.
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 $\frac{2π}{p_1},\frac{2π}{p_2},\dots,\frac{2π}{p_n}$. 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 $n \geq 2$.