Example of an amenable finitely generated and presented group with a non-finitely generated subgroup - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:34:30Z http://mathoverflow.net/feeds/question/65045 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65045/example-of-an-amenable-finitely-generated-and-presented-group-with-a-non-finitely Example of an amenable finitely generated and presented group with a non-finitely generated subgroup Alessandro Carderi 2011-05-15T15:18:22Z 2011-05-15T20:43:03Z <p>I'm looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated.</p> <p>The question is easy for finitely generated amenable group and an example is the lamp-lighter group $C_2\wr \mathbb{Z}$. </p> <p>An Abelian and finitely generated group has no such subgroups. There exists a bigger class of groups with this property? </p> http://mathoverflow.net/questions/65045/example-of-an-amenable-finitely-generated-and-presented-group-with-a-non-finitely/65052#65052 Answer by Mark Sapir for Example of an amenable finitely generated and presented group with a non-finitely generated subgroup Mark Sapir 2011-05-15T18:11:10Z 2011-05-15T18:11:10Z <p>There are finitely presented metabelian groups containing the lamplighter groups. One of them was constructed by Baumslag: $\langle a,b,c \mid a^2=1, [b,c]=1, [a^b,a]=1, a^c=a^ba\rangle$.</p> http://mathoverflow.net/questions/65045/example-of-an-amenable-finitely-generated-and-presented-group-with-a-non-finitely/65053#65053 Answer by Derek Holt for Example of an amenable finitely generated and presented group with a non-finitely generated subgroup Derek Holt 2011-05-15T18:17:29Z 2011-05-15T18:17:29Z <p>I don't know much about amenable groups I am afraid, but according to the Wikipedia article, all solvable groups are amenable. So we can take the Baumslag-Solitar group</p> <p>$B(1,n) = \langle x,y \mid y^{-1}xy = x^n \rangle.$</p> <p>If we let $N$ be the normal closure of the subgroup generated by $x$, then $N$ is abelian with $G/N$ cyclic, but $N$ is not finitely generated when $n > 1$. Note also that $B(1,n)$ is isomorphic to the subgroup of ${\rm GL}(2, \mathbb{Q})$ generated by</p> <p>$x = \left(\begin{array}{cc}1&amp;0\\1&amp;1\end{array}\right)$ and $y = \left(\begin{array}{cc}n&amp;0\\0&amp;1\end{array}\right).$</p> http://mathoverflow.net/questions/65045/example-of-an-amenable-finitely-generated-and-presented-group-with-a-non-finitely/65069#65069 Answer by Alain Valette for Example of an amenable finitely generated and presented group with a non-finitely generated subgroup Alain Valette 2011-05-15T20:43:03Z 2011-05-15T20:43:03Z <p>By the way, you may enjoy the fact, due to G. Baumslag, that a standard wreath product $W\wr G$ with $W\neq 1$ and $G$ infinite, is never finitely presented; see Gilbert Baumslag. Wreath products of finitely presented groups. Math. Z. 75 , 22-28, 1961. For finite presentability of permutational wreath products, see a paper by Cornulier: <a href="http://www.normalesup.org/~cornulier/wrea_fp.pdf" rel="nofollow">http://www.normalesup.org/~cornulier/wrea_fp.pdf</a></p>