User rostyslav kravchenko - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:27:44Z http://mathoverflow.net/feeds/user/10714 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54401/fundamental-group-of-a-thick-part-of-hyperbolic-manifold Fundamental group of a thick part of hyperbolic manifold Rostyslav Kravchenko 2011-02-05T14:32:11Z 2011-02-05T21:15:51Z <p>Let $M$ be a complete hyperbolic manifold of dimension $n$, let $\varepsilon=\varepsilon_n$ be the Margulis constant. Let $M_{[\varepsilon,\infty)}$ be the thick part of $M$ with respect to $\varepsilon$. Suppose that $\pi_1(M)$ is infinite. Is it true that $\pi_1(M_{[\varepsilon,\infty)})$ is also infinite. Or, in case $M_{[\varepsilon,\infty)}$ is not connected, whether there exist a connected component of $M_{[\varepsilon,\infty)}$ such that its fundamental group is infinite.</p> <p>PS: I am reading a paper, where this fact seems to be an important point in the proof. This is totally not my area, so it might as well be trivial for anyone familiar with these notions, however, I will be much obliged for an expalnation why this is (or is not) true.</p> http://mathoverflow.net/questions/52509/subspace-of-l2-that-lies-in-l-infty Subspace of $L^2$ that lies in $L^\infty$ Rostyslav Kravchenko 2011-01-19T12:50:28Z 2011-01-19T19:08:13Z <p>Let $E$ be a closed subspace of $L^2[0,1]$. Suppose that $E\subset{}L^\infty[0,1]$. Is it true that $E$ is finite dimensional?</p> <p>PS. This is actually a question from the real analysis qualifier. I came across it as I was teaching qualifier preparation course, and was solving problems from old qualifiers. So, though it might follow from some advanced theory of Banach spaces, I am most interested in the 'elementary' solution, using only methods from standard real analysis course. Note: if $E\subset{}C[0,1]$, then it is a problem from Folland, and there is a solution there. However, it does not work for $L^\infty$, not without some trick.</p> http://mathoverflow.net/questions/54401/fundamental-group-of-a-thick-part-of-hyperbolic-manifold/54418#54418 Comment by Rostyslav Kravchenko Rostyslav Kravchenko 2011-02-05T18:28:51Z 2011-02-05T18:28:51Z Thank you, that settles it for $n&gt;2$ for me, since then $M_{[\varepsilon,\infty)}$ is connected, as I understand. Is it still connected if $n=2$? (Why for instance a short curve cannot separate 2 components?) http://mathoverflow.net/questions/52509/subspace-of-l2-that-lies-in-l-infty/52525#52525 Comment by Rostyslav Kravchenko Rostyslav Kravchenko 2011-01-19T18:44:22Z 2011-01-19T18:44:22Z Many thanks! This is actually the trick to extend the solution from Folland from $C[0,1]$ to $L^\infty$.