The Salem conjecture Sample Clauses

The Salem conjecture is equivalent to the Short Geodesic conjecture for arithmetic hyperbolic 3-orbifolds with invariant trace field k = k¯. The Salem conjecture says that certain roots of integral polynomials are uniformly bounded below by a number strictly greater than 1. The Short Geodesic conjecture says that the length of closed geodesics are uniformly bounded below. Both con- jectures will be explicitly mentioned in Section 6.6. Finally, in Section 7, we have compiled results that were proven along the way that do not quite line up with the main thrust of the paper. However, they answer some interesting questions and they are included as well. Now that the results of the paper have been enumerated, we need to outline what background material is necessary to prove these results. In sections 2.1, 2.2, 3, and 4.1, we will introduce the necessary background material from hyperbolic 3-space, hy- perbolic 3-manifolds, algebraic number theory, and arithmetic hyperbolic 3-manifolds. After that, in Section 5, we will review some pertinent results that set the stage for our investigation.

The Salem conjecture is equivalent to the Short Geodesic conjecture for arithmetic hyperbolic 3-orbifolds with invariant trace field k = k¯. The conjectures are now listed. Conjecture 6.34 (Salem conjecture) There exists ms > 1 such that if u is a Salem number, then u ≥ ms. Conjecture 6.35 (Short Geodesic conjecture for dimension 3 with k =