Margulis Numbers and Number Fields

Let M be a closed, orientable hyperbolic 3-manifold such that (1) H1(M ; Zs) = 0 for s = 2, 3, 7. Suppose that (2) M is non-Haken, or more generally that it has integral traces. If the trace field of M is quadratic then 0.395 is a Margulis number for M . If the trace field is cubic then 0.3 is a Margulis number for M . If K is any number field, then for all but finitely many closed, orientable hyperbolic 3-manifolds M which satisfy (1) and (2) and have trace field K, the number 0.183 is a Margulis number for M . Furthermore, if K is any number field, there is a real number with 0 < ≤ 0.3, having the following property. Let M be any closed hyperbolic 3-manifold which satisfies (1) and (2) and has trace field K. Then about every primitive closed geodesic in M having length l < there is an embedded tube having radius R(l), where R(l) is an explicitly defined function such that sinhR(l) is asymptotic to (.01869...)/l.