TITLE: Best known Margulis constants?
QUESTION [12 upvotes]: 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?

REPLY [3 votes]: 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.