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In the literature, there are several examples of solvable groups acting faithfully by order-preserving homeomorphisms of the real line. There are also examples of groups of intermediate growth with such actions (see for instance A. Navas "Groups of circle diffeomorphisms" arXiv). So my question is:

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

Where by the class of subexponentially-amenable groups 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 faithful action by order preserving homeomorphisms 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?

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  • $\begingroup$ Does the Grigorchuk group act in this way? $\endgroup$
    – Noah Schweber
    Commented Jan 7, 2014 at 21:01
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    $\begingroup$ Finitely generated elementary amenable groups cannot have intermediate growth (C. Chow, 1980). Hence, examples of intermediate growth answer your question. $\endgroup$
    – Misha
    Commented Jan 7, 2014 at 21:32
  • $\begingroup$ 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. $\endgroup$
    – Misha
    Commented Jan 7, 2014 at 22:10
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    $\begingroup$ @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). $\endgroup$
    – Cristobal Rivas
    Commented Jan 9, 2014 at 14:02
  • $\begingroup$ @Misha I have edited the Question, thank you both! $\endgroup$
    – Cristobal Rivas
    Commented Jan 9, 2014 at 14:04

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