The growth of torus link groups

TL;DR

This paper explicitly computes the rational growth functions of (p,q)-torus link groups for all p, q > 1 and demonstrates that their growth rates are Perron numbers, contributing to the understanding of their algebraic properties.

Contribution

It provides explicit formulas for the rational growth functions of (p,q)-torus link groups, a new result in the study of their algebraic and geometric properties.

Findings

Growth functions are rational for all p, q > 1.

Growth rates are Perron numbers.

Explicit formulas for the growth functions are derived.

Abstract

Let $G$ be a finitely generated group with a finite generating set $S$. For $g\in G$, let $l_S(g)$ be the length of the shortest word over $S$ representing $g$. The growth series of $G$ with respect to $S$ is the series $A(t) = \sum_{n=0}^\infty a_n t^n$, where $a_n$ is the number of elements of $G$ with $l_S(g)=n$. If $A(t)$ can be expressed as a rational function of $t$, then $G$ is said to have a rational growth function. We calculate explicitly the rational growth functions of $(p,q)$-torus link groups for any $p, q > 1.$ As an application, we show that their growth rates are Perron numbers.