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October 2018 A new formula for the spherical growth series of an amalgamated free product of two infinite cyclic groups
Michihiko Fujii
Kodai Math. J. 41(3): 475-511 (October 2018). DOI: 10.2996/kmj/1540951250

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.

Citation

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Michihiko Fujii. "A new formula for the spherical growth series of an amalgamated free product of two infinite cyclic groups." Kodai Math. J. 41 (3) 475 - 511, October 2018. https://doi.org/10.2996/kmj/1540951250

Information

Published: October 2018
First available in Project Euclid: 31 October 2018

zbMATH: 07000580
MathSciNet: MR3870700
Digital Object Identifier: 10.2996/kmj/1540951250

Rights: Copyright © 2018 Tokyo Institute of Technology, Department of Mathematics

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Vol.41 • No. 3 • October 2018
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