More on loxodromic eigenvalues Sample Clauses

More on loxodromic eigenvalues. Remark 7.1 The norm of a loxodromic eigenvalue does not completely determine the angle of the eigenvalue as seen above. But, the angle of a generic loxodromic eigenvalue completely determines the norm of the eigenvalue. Since we know the conjugates of a loxodromic eigenvalue, we can determine the possible conjugates of λ/λ¯ = e2iθ. The possible conjugates have one of the following forms: r2, λ· eiφ, λ¯ · eiφ, and eiφ1 · eiφ2 . (Of course, there are more possibilities. We could take reciprocals and complex-conjugate if we wanted.) Either e2iθ only has conjugates of norm 1, in which case it is a root of unity, or it has at least one conjugate of norm r2 or r. The latter case means that eiθ has a real conjugate of norm r or a non-real conjugate of norm √r, and, thus, the angle completely determines the norm of the eigenvalue because any other norm that occurs will be 1. We have pretty much pinned down what loxodromic elements may occur in a quadratic extension of a trace field k =ƒ k¯. That is, the only loxodromic eigenvalues that do occur are the ones we have mentioned thus far. This is summed up in Proposition 7.3 with the help of Lemma 7.2, which we state and prove now.