Mathematics – Differential Geometry
Scientific paper
2009-02-06
Mathematics
Differential Geometry
This is a completely new paper. Many of the results of my previously posted paper of the same title were subsumed by my paper
Scientific paper
It is shown that, up to isometry, all but finitely many closed, orientable hyperbolic 3-manifolds with a given trace field $K$ admit 0.34 as a Margulis number. This is deduced from a more technical result giving a condition under which $\max(d(P,x\cdot P),d(P,y\cdot P))\ge0.34$ for every $P\in\HH^3$, where $x$ and $y$ lie in $\pizzle(E)$ for some number field $E$, generate a discrete torsion-free group of $\pizzle(\CC)$ and do not commute. Specifically, this is always the case if there is a valuation $v$ of $E$ such that (1) the residue field $k_v=\frako_v/\frakm_v$ of $v$ has sufficiently large characteristic, (2) $x\in\pizzle(\frako_v)$, and (3) the image of $x$ under the natural homomorphism $\pizzle(\frako_v)\to \pizzle(k_v)$ has order 7.
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