Irreducible components in an algebraic variety of representations of a family of one-relator groups

Abstract

Given a finitely generated group G, the set Hom(G,SL_2 C) inherits the structure of an algebraic variety R(G)called the "representation variety" of G. This algebraic variety is an invariant of G. Let G_{pt}=< a, b; a^p= b^t>, where p, t are integers greater than one. In this paper a formula is produced yielding the number of four dimensional irreducible components of the affine algebraic variety R(G_{pt}). A direct consequence of the main theorem of this paper is that if K is a torus knot, then its genus equals the number of four dimensional components of R(G_{pt}) corresponding to its knot group G_{pt}.