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

Let us consider the group G = 〈 x , y | x m = y n 〉 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 = t x , Y = t y and Z = t x y . 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 ) .