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.](http://www.ams.org/mathscinet/search/publdoc.html?extend=1&pg1=IID&s1=224643&vfpref=html&r=18&mx-pid=952231))


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](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=shalen&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=2866926))


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?



Best known Margulis constants?
For question 2, the best known is due to Ruth Kellerhals (the answer is [here](http://dl.dropbox.com/u/5188175/2106991.pdf))


And for question 1, the latest (but, judging from [the math review](http://dl.dropbox.com/u/5188175/2175884.pdf), not greatest) is Gehring/Martin.


