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?