Margulis numbers and number fields

TL;DR

The paper proves that most hyperbolic 3-manifolds with a fixed trace field have a Margulis number of at least 0.34, based on algebraic conditions involving number fields and valuations.

Contribution

It establishes a new criterion involving valuations of number fields that guarantees a uniform Margulis number for hyperbolic 3-manifolds with a given trace field.

Findings

Most hyperbolic 3-manifolds with a fixed trace field have Margulis number ≥ 0.34.

A technical condition involving valuations and residue fields ensures this Margulis number.

The result applies to manifolds with non-commuting elements in certain algebraic groups.

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 $\max(d(P,x\cdot P),d(P,y\cdot P))\ge0.34$ 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.