On a Theorem of Burde and de Rham

We generalize a theorem of Burde and de Rham characterizing the zeros of the Alexander polynomial. Given a representation of a knot group π, we define an extension π̃ of π, the Crowell group. For any GLNC representation of π, the zeros of the associated twisted Alexander polynomial correspond to representations of π̃ into the group of dilations of C . A classic theorem of G. Burde, and independently G. de Rham characterizes the zeros of the Alexander polynomial of a knot in terms of representations of the knot group by dilations, certain affine transformations of the complex plane. More recently, twisted Alexander polynomials, which incorporate information from linear representations of the knot group, have proved useful in many areas (see the survey [5]). We generalize the theorem of Burde and de Rham for twisted Alexander polynomials, replacing dilations of the complex plane with dilations of C , where N is the dimension of the representation. The knot group is replaced by a natural extension, the Crowell group, introduced in [4]. 1 Theorem of Burde and deRham Definition 1.1. A dilation of C is a transformation d : C → C of the form d(z) = αz + b, for some α ∈ C = C \ {0} and b ∈ C . The number α is called the dilation ratio of d. Dilations of C form a group DN under composition. It is easy to see that conjugate elements have the same dilation ratio α. ∗Both authors partially supported by NSF grant DMS-0706798.