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?