Combinatorial aspects of the character variety of a family of one-relator groups

Let us consider the group G=〈x,y|xm=yn〉 with m and n nonzero integers. In this paper, we study 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=tx, Y=ty and Z=txy. 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. Finally 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).