## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 41, Number 3 (2018), 475-511.

### 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 $G(p,q) = \langle x, y|x^p = y^q\rangle$, with integers $p$ and $q$ satisfying $2 \leq p \leq q$. 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 $\frac{2\pi}{p}$ and $\frac{2\pi}{q}$. 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.

#### Article information

**Source**

Kodai Math. J., Volume 41, Number 3 (2018), 475-511.

**Dates**

First available in Project Euclid: 31 October 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1540951250

**Digital Object Identifier**

doi:10.2996/kmj/1540951250

**Mathematical Reviews number (MathSciNet)**

MR3870700

**Zentralblatt MATH identifier**

07000580

#### Citation

Fujii, Michihiko. A new formula for the spherical growth series of an amalgamated free product of two infinite cyclic groups. Kodai Math. J. 41 (2018), no. 3, 475--511. doi:10.2996/kmj/1540951250. https://projecteuclid.org/euclid.kmj/1540951250