On the varieties of representations and characters of a family of one-relator subgroups. Their irreducible components

Let us consider the group $G = < x,y \mid x^m = y^n>$ with $m$ and $n$ nonzero integers. In this paper, we study the variety of epresentations $R(G)$ and the character variety $X(G)$ in $SL(2,\C)$ of the group $G$,obtaining by elementary methods an explicit primary decomposition of the ideal corresponding to $X(G)$ in the coordinates $X=t_x$, $Y=t_y$ and $Z=t_{xy}$. As an easy consequence, a formula for computing the number of irreducible components of $X(G)$ as a function of $m$ and $n$ is given. We provide a combinatorial description of $X(G)$ and we prove that in most cases it is possible to recover $(m,n)$ from the combinatorial structure of $X(G)$. Finally we compute the number of irreducible components of $R(G)$ and study the behavior of the projection $t:R(G)\longrightarrow X(G)$.