# Examples of amenable groups acting on the real line

In the literature, there are several examples of solvable groups acting by order preserving homeomorphisms of the real line. There are also examples of groups of intermediate growth acting in the same way (see for instance A. Navas "Groups of Circle Diffeomorphhism"). So my question is:

Is there an example of an amenable but not subexponentially-amenable group acting by order preserving homeomorpism of the real line?

Where by the class of subexponentially-amenable group I mean a the smallest class containing the groups of subexponential growth and which is closed under taking subgroups, quotients, extensions and direct limits.

I should also point out that, for countable groups, having a faithfull action by order preserving homeos of the real line is equivalent to admitting a left-ordering, that is a total ordering which is invariant under left multiplication. So an equivalent question is

Is there an example of a left-orderable amenable group that is not subexponentially-amenable?

• Does the Grigorchuk group act in this way? – Noah Schweber Jan 7 '14 at 21:01
• Finitely generated elementary amenable groups cannot have intermediate growth (C. Chow, 1980). Hence, examples of intermediate growth answer your question. – Misha Jan 7 '14 at 21:32
• You also have to clarify what you mean by "non-elementary amenable", since it can be interpreted as either (1) amenable group which is not elementary amenable, (2) group which fails to be elementary amenable. In the latter case, examples are much easier since one can easily construct a free nonabelian group of diffeomorphisms of line preserving orientation. – Misha Jan 7 '14 at 22:10
• @Noah The Grigorchuk group have torsion, so it does not acts by order preserving homeos of the real line. However, there is a variation of it, from Grigorchuk and Maki, which acts on the real line (and still have intermediate growth). – Cristobal Rivas Jan 9 '14 at 14:02
• @Misha I have edited the Question, thank you both! – Cristobal Rivas Jan 9 '14 at 14:04