6
$\begingroup$

For a given class of groups, called base class, $B$. Osin have defined elementary closure of $B$ as the minimal class that contains $B$ and is closed under taking subgroups, extensions, direct unions, quotients. Previously, this have been considered by Day, von Neumann, Chou, who took $B$ to be the class of all abelian and finite groups. In this case, the elementary closure of $B$ is called elementary amenable groups. One can also take $B$ as the class of all groups with subexponential growth.

An extra condition on $B$, which simplifies computations related to the elementary closure is what Osin calls SQ-closed, i.e., we assume that $B$ is closed under taking quotients and subgroups.

I wonder what are other natural classes of amenable groups $B$ to consider? (Of course, I don't mean obvious classes like "all finite" or "all abelian".)

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ The class you obtain from abelian groups is not obvious: for instance, it contains non-solvable finitely generated groups, but not all elementary amenable groups (since it misses nonabelian finite simple groups). Possibly you want classes larger than the class of amenable groups? $\endgroup$
    – YCor
    Aug 24, 2016 at 0:53
  • $\begingroup$ Uniformly amenable groups seem to be a natural class. These were introduced by Keller and Bozejko. projecteuclid.org/euclid.ijm/1256052282 link.springer.com/article/10.1007%2FBF01420276 $\endgroup$
    – Ian Agol
    Aug 24, 2016 at 19:40
  • $\begingroup$ Yves, you are right, I mean the classes that are more then elementary amenable groups. Ian, thanks, good point. This seems to be an interesting question, maybe some kind of dynamics can define this, something like groups acting on trees. $\endgroup$
    – Kate Juschenko
    Aug 26, 2016 at 0:33
  • $\begingroup$ Yves, any citation to examples of non-solvable finitely generated groups which are not in EG(Abelian)? $\endgroup$
    – Kate Juschenko
    Aug 26, 2016 at 0:34

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.