Examples of Amenable Groups other than Z_n

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.

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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

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.

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

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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.

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Groups generated by bounded automaton are amenable:

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

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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

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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)

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