Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I'm reading about amenable groups. What are explicit examples of nonabelian discrete amenable groups other than finite groups? Perhaps a group presentation or matrix representation would be useful.

share|improve this question
7  
Shouldn't this question be community-wiki, since the goal is to have a list rather than to find a definitive answer? –  Yemon Choi Apr 20 '11 at 1:02

9 Answers 9

To "specify" Alain's answer: 0) The group $\langle a,b \mid bab^{-1}=a^2\rangle$ (solvable of class 2 Baumslag-Solitar group) 1) The group of upper triangular $n\times n$ matrices with integer coefficients and 1 on the diagonal (nilpotent), $n\ge 1$. 2) The group of all permutations of $\mathbb{Z}$ with finite support (locally finite). 3) The subexp. growth groups, unfortunately, would require more space to define. But you can find them in Wiki.

share|improve this answer

Solvable (which includes nilpotent and polycyclic); locally finite; subexponential growth.

share|improve this answer
1  
can you be more explicit? –  john mangual Apr 19 '11 at 18:09

The lamplighter group, defined as the wreath product $\mathbb{Z}/2\mathbb{Z} \wr \mathbb{Z}$, is amenable yet has exponential growth. It can be thought of as a bi-infinite sequence of street lamps, each of which can be turned on and off, and a lamplighter who moves along the sequence. The three generators of the group are to move the lamplighter right or left, and to switch the state of the lamp he is positioned in front of. With this picture in mind, it is easy to show the group has exponential growth.

share|improve this answer

Groups generated by bounded automaton are amenable:

http://dx.doi.org/10.1215/00127094-2010-046

share|improve this answer
    
Mustafa, the link is broken –  Kate Juschenko Apr 19 '11 at 19:45
    
Thanks Kate i fixed it. –  Mustafa Gokhan Benli Apr 19 '11 at 22:15

$\mathrm{Symm}(\mathbb{Z}) \leftthreetimes \mathbb{Z}$ - see page 4321 in http://www.cse.sc.edu/~maxal/a-g-g.pdf

share|improve this answer

Elementary amenable groups: the smallest class of groups which includes finite groups and commutative groups, and is closed under formation of homomorphic images, subgroups, group extensions (by an other element of the class) and directed unions. (See http://en.wikipedia.org/wiki/Elementary_amenable_group)

share|improve this answer

Solvable and locally finite are elementary, I guess those of subexponential growth are, too.

share|improve this answer

E.g. a solvable group is one that can be constructed from abelian groups using extensions. But now I see that locally finite groups are not elementary (for example, the free group on two generators is finitely generated, hence locally finite, but it is definitely not amenable, hence it cannot be elementary), sorry for that.

share|improve this answer
    
Free groups are not locally finite. I think you meant that inverse limit of finite groups need not be amenable. –  Misha Apr 4 '13 at 20:59
    
free groups are not inverse limits of finite groups neither (at least not in the category of groups). They are residually finite, anyway (that is, in the category of marked groups, the same as inverse limits of finite groups). –  YCor Apr 4 '13 at 22:01
    
Yves, I meant that they embed in inverse limits of finite groups. –  Misha Apr 4 '13 at 23:03

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.