Margulis numbers and number fields

Abstract

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 $\max (d(P,x\cdot P),d(P,y\cdot P))\ge 0.34$ 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.

This is a completely new paper. Many of the results of my previously posted paper of the same title were subsumed by my paper "A generic Margulis number for hyperbolic 3-manifolds." In the new paper I get a considerably stronger generic Margulis number for the class of manifolds with a prescribed trace field than for the class of all (closed, orientable hyperbolic 3-)manifolds. (54 pages)

References in corpus (5)

• Tameness of hyperbolic 3-manifolds
• Dehn surgery, homology and hyperbolic volume
• Systoles of hyperbolic 4-manifolds
• Small optimal Margulis numbers force upper volume bounds
• Margulis numbers for Haken manifolds