Example of an amenable finitely generated and presented group with a non-finitely generated subgroup

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

I'm looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated.

The question is easy for finitely generated amenable group and an example is the lamp-lighter group $C_2\wr \mathbb{Z}$.

An Abelian and finitely generated group has no such subgroups. There exists a bigger class of groups with this property?

-

3 Answers

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

$B(1,n) = \langle x,y \mid y^{-1}xy = x^n \rangle.$

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

$x = \left(\begin{array}{cc}1&0\\1&1\end{array}\right)$ and $y = \left(\begin{array}{cc}n&0\\0&1\end{array}\right).$

-

2
The subgroup generated by $x$ is not normal! – Steve D May 15 2011 at 20:08

1
But the normal subgroup generated by $x$ is normal and infinitely generated. It is isomorphic to the additive group of diadic rational numbers. – Mark Sapir May 16 2011 at 5:59

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.

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: http://www.normalesup.org/~cornulier/wrea_fp.pdf

-

There are finitely presented metabelian groups containing the lamplighter groups. One of them was constructed by Baumslag: $\langle a,b,c \mid a^2=1, [b,c]=1, [a^b,a]=1, a^c=a^ba\rangle$.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 18, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9337502717971802, "perplexity_flag": "head"}