Algebraic geometric invariants for a class of one-relator groups

Abstract

Given a finitely generated group H, the set Hom( H, SL 2 C) inherits the structure of an affine algebraic variety R( H) called the representation variety of H. Let a one-relator group with presentation G = 〈x 1, …, x n, y; W( x ̄ ) = y k〉 be given, where W( x ̄ ) ≠ 1 is in the free group on the generators { x ̄ } = {x 1, …, x n} , and k ≥ 2. In this paper a theorem will be proven allowing the computation of Dim( R( G)) in terms of subvarieties of the representation variety of the free group on n generators, R( F n ), arising from solutions to the equation W( x ̄ ) = ± l in SL 2 C. Conditions are given guaranteeing the reducibility of R( G). Finally, applications to the class of one-relator groups with non-trivial center are made.