Universal group? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:01:52Z http://mathoverflow.net/feeds/question/19895 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19895/universal-group Universal group? Anton Petrunin 2010-03-30T23:30:31Z 2012-03-22T19:03:31Z <p>I can construct a finitely presented group $G$ with the following property (which I use to construct something else).</p> <blockquote> <p>Given a finitely preseted group $\Gamma$, there is a subgroup $G'\le G$ of finite index such that $$\Gamma=G'/\langle\mathrm{Tor}\, G'\rangle ,$$ where $\mathrm{Tor}\, G'\subset G'$ is the set of all elements of finite order. </p> </blockquote> <p>I think to call such group $G$ <em>universal</em>.</p> <p><strong>Questions:</strong></p> <ul> <li>Was it already constructed? </li> <li>Does it already has a name? Is there any closely related terminology?</li> </ul> <hr> <p><strong>P.S.</strong> </p> <ul> <li>The group which I construct is in fact hyperbolic.</li> <li>My construction is simple, but it takes 2--3 pages. Let me know if you see a short way to do it.</li> <li><a href="http://www.springerlink.com/content/k332884x7m10l654/" rel="nofollow">Here</a>, the term "universal group" was used in very similar context (thanks to D. Panov for the reference).</li> <li>Thanks to all your comments, we call them <a href="http://front.math.ucdavis.edu/1104.4814" rel="nofollow">"all-inclusive" actions</a> now.</li> </ul> http://mathoverflow.net/questions/19895/universal-group/77575#77575 Answer by Anton Petrunin for Universal group? Anton Petrunin 2011-10-09T02:48:47Z 2011-10-09T02:48:47Z <p>I was asked write in an answer to move the question to answered status.</p> <p>Thank you all for your comments they were helpful for me and Dima.</p>