Published online by Cambridge University Press: 11 February 2015
We prove that if ${\rm\Gamma}$ is a subgroup of
$\text{Diff}_{+}^{\,1+{\it\epsilon}}(I)$ and
$N$ is a natural number such that every non-identity element of
${\rm\Gamma}$ has at most
$N$ fixed points, then
${\rm\Gamma}$ is solvable. If in addition
${\rm\Gamma}$ is a subgroup of
$\text{Diff}_{+}^{\,2}(I)$, then we can claim that
${\rm\Gamma}$ is meta-abelian.