Let $G$ be a finitely generated group with a finite generating set $S$. For $gin G$, 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.
We discuss a problem posed by Gersten: Is every automatic group which does not contain Z+Z subgroup, hyperbolic? To study this question, we define the notion of n-tracks of length n, which is a structure like Z+Z, and prove its existence in the non-h
yperbolic automatic groups with mild conditions. As an application, we show that if a group acts effectively, cellularly, properly discontinuously and cocompactly on a CAT(0) cube complex and its quotient is weakly special, then the above question is answered affirmatively.
