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A Margulis number for a hyperbolic $n$-manifold $M=\mathbb{H}^n/\Gamma$ is a number $\epsilon>0$ such that for each $x\in\mathbb{H}^n$ the group generated by the elements in $\Gamma$ which move $x$ less than distance $\epsilon$ is elementary. The Margulis constant for hyperbolic $n$-manifolds is the largest number $\epsilon(n)$ which is a Margulis number for every hyperbolic $n$-manifold.

Meyerhoff showed that $\epsilon(3) > 0.104$. (Robert Meyerhoff. A lower bound for the volume of hyperbolic 3-manifolds. Canad. J. Math., 39(5):1038–1056, 1987.)

Shalen proved that 0.29 is a Margulis number for all but finitely many orientable hyperbolic 3-manifolds. He also notes that experimental evidence suggests that $\epsilon(3) <0.616$ (Peter Shalen. Topology and geometry in dimension three, 103–109, Contemp. Math., 560, Amer. Math. Soc., Providence, RI, 2011)

Question 1: Is Meyerhoff's lower bound the best known lower bound for $\epsilon(3)$?

Question 2: What is known about the Margulis constants for higher dimensional hyperbolic manifolds?

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3 Answers 3

up vote 6 down vote accepted

For question 2, the best known is due to Ruth Kellerhals (the answer is here)

And for question 1, the latest (but, judging from the math review, not greatest) is Gehring/Martin.

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Results of Culler and Shalen together with tameness, density, etc. imply that there exists a number $V$ such that if $M$ is a hyperbolic 3-manifold of volume $>V$, then the Margulis constant of $M$ is $\geq \log(3)$. This indicates that one ought to be able to compute the Margulis constant, making it an a priori trivial problem. To prove this, you take a sequence of 2-generator groups realizing the Margulis constant for manifolds with volume approaching $\infty$. In the limit, the Margulis constant is $>\log(3)$ by Culler-Shalen, so one concludes that there is some bound on volume for manifolds with Margulis constant $<\log(3)$.

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For Question 2: In addition to Kellerhals' lower bounds on $\epsilon(n)$, there exists an absolute constant $C>0$ such that $$ \epsilon(n)\le \frac{C}{\sqrt{n}}, $$ see Proposition 5.2 here.

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