Best known Margulis constants? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:44:52Z http://mathoverflow.net/feeds/question/92295 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92295/best-known-margulis-constants Best known Margulis constants? bb 2012-03-26T17:43:26Z 2012-04-12T03:38:10Z <p>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.</p> <p>Meyerhoff showed that $\epsilon(3) > 0.104$. (<a href="http://www.ams.org/mathscinet/search/publdoc.html?extend=1&amp;pg1=IID&amp;s1=224643&amp;vfpref=html&amp;r=18&amp;mx-pid=952231" rel="nofollow">Robert Meyerhoff. A lower bound for the volume of hyperbolic 3-manifolds. Canad. J. Math., 39(5):1038–1056, 1987.</a>)</p> <p>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 $\epsilon(3) &lt;0.616$ (<a href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&amp;co4=AND&amp;co5=AND&amp;co6=AND&amp;co7=AND&amp;dr=all&amp;pg4=AUCN&amp;pg5=TI&amp;pg6=PC&amp;pg7=ALLF&amp;pg8=ET&amp;review_format=html&amp;s4=shalen&amp;s5=&amp;s6=&amp;s7=&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq&amp;r=1&amp;mx-pid=2866926" rel="nofollow">Peter Shalen. Topology and geometry in dimension three, 103–109, Contemp. Math., 560, Amer. Math. Soc., Providence, RI, 2011</a>)</p> <p>Question 1: Is Meyerhoff's lower bound the best known lower bound for $\epsilon(3)$?</p> <p>Question 2: What is known about the Margulis constants for higher dimensional hyperbolic manifolds?</p> http://mathoverflow.net/questions/92295/best-known-margulis-constants/92309#92309 Answer by Igor Rivin for Best known Margulis constants? Igor Rivin 2012-03-26T20:56:14Z 2012-03-26T21:02:27Z <p>For question 2, the best known is due to Ruth Kellerhals (the answer is <a href="http://dl.dropbox.com/u/5188175/2106991.pdf" rel="nofollow">here</a>)</p> <p>And for question 1, the latest (but, judging from <a href="http://dl.dropbox.com/u/5188175/2175884.pdf" rel="nofollow">the math review</a>, not greatest) is Gehring/Martin.</p> http://mathoverflow.net/questions/92295/best-known-margulis-constants/93819#93819 Answer by Agol for Best known Margulis constants? Agol 2012-04-12T03:38:10Z 2012-04-12T03:38:10Z <p>Results of <a href="http://www.ams.org/mathscinet-getitem?mr=1135928" rel="nofollow">Culler and Shalen</a> 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 $&lt;\log(3)$. </p>