How to determine free generators of a closed subgroup of a free pro-\$p\$-group ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T12:23:43Z http://mathoverflow.net/feeds/question/83005 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83005/how-to-determine-free-generators-of-a-closed-subgroup-of-a-free-pro-p-group How to determine free generators of a closed subgroup of a free pro-\$p\$-group ? Joël 2011-12-08T21:25:29Z 2011-12-09T04:24:43Z <p>If \$F\$ is a free discrete group, then any subgroup \$H\$ of \$F\$ is free: this is the well-known theorem of Nielsen-Schreier. Moreover, there is a well-known algorithm, the <em>Nielsen-Schreier method</em> that allows us to determine a set of free generators of \$H\$ (see e.g. <a href="http://books.google.com/books?id=1D6crOEoRFEC&amp;pg=PA12&amp;lpg=PA12&amp;dq=explicit+nielsen+schreier&amp;source=bl&amp;ots=d0gff7dXlZ&amp;sig=Hx-dOYEklHpRvR15pj8Lzcn2lAQ&amp;hl=en&amp;ei=gHPfTtq_IYbt0gGc9ZSzBw&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CCMQ6AEwAA#v=onepage&amp;q=explicit%2520nielsen%2520schreier&amp;f=false" rel="nofollow">this reference</a>). </p> <p>Now if \$F\$ is free pro-\$p\$-group, then any closed subgroup \$H\$ of \$F\$ is pro-\$p\$-free (cf Ribes and Zaleskii, Profinite groups, Cor. 7.7.5). But is there a method to describe a space on which \$H\$ is free?</p> <p>I am interested in the question even in the perhaps much simpler case where \$F\$ is the pro-p-completion of a free discrete group \$F_0\$ (with finitely many generators), and \$H\$ is the closure in \$F\$ of a subgroup \$H_0\$ of \$F_0\$ (of which I happen to know a set of free generators --infinitely many-- explicitly by the Nielsen-Schreier methods). Then by right-exacness of the \$p\$-completion functor, \$H\$ is a quotient of the completion of \$H_0\$, but I can't see by what (if anything). </p> <p>This question is a follow-up of <a href="http://mathoverflow.net/questions/82847/a-metabelian-quotient-of-a-free-group" rel="nofollow">this one</a>.</p> http://mathoverflow.net/questions/83005/how-to-determine-free-generators-of-a-closed-subgroup-of-a-free-pro-p-group/83011#83011 Answer by Benjamin Steinberg for How to determine free generators of a closed subgroup of a free pro-\$p\$-group ? Benjamin Steinberg 2011-12-08T23:35:23Z 2011-12-08T23:35:23Z <p>This doesn't quite answer your question, but... If \$F\$ is a discrete free group and \$H\$ is a fg subgroup, Ribes and Zalesskii gave an algorithm, improved by Margolis, Sapir and Weil, to compute the pro-p closure K of H in F. In this case K is fg and its closure in the pro-p completion of F is its pro-p completion. I am not sure when H is not fg what happens. </p>