On the rings of Fricke characters of free abelian groups

Abstract

In this paper, we determine the structure of the rings $\mathfrak{X}_{\Q}(H)$ of Fricke characters over $\Q$ of free abelian groups $H$ of rank $n \geq 1$. In particular, we consider the ideal $J$ in $\mathfrak{X}_{\Q}(H)$, generated by $\mbox{tr\,}x -2$ for any $x \in H$ and give a $\Q$-basis of each of the graded quotients $\mbox{gr}^k(J):=J^k/J^{k+1}$ for $k \geq 1$. Then we introduce a weight for each element of $\mathfrak{X}_{\Q}(H)$. By using the concept of this weight we show that $\mathfrak{X}_{\Q}(H)$ is an integral domain.