Existence of an orbit of exponential growth for group acting on the real line

Question

Let G be a non-abelian finitely generated subgroup of increasing homeomorphisms of the real line having a fixed point free element $h$ ($hx>x$ for all $x$ in the line). Is there a real number $a$ such that its orbit $Ga = \{ga: g\in G\}$ does not have a sub-exponential growth?

Answer 1

No, the hypotheses do not imply the conclusion. For example, every nontrivial, finitely generated, torsion-free, nilpotent group has an action of the type you describe, but every orbit has polynomial growth (because the group has polynomial growth).