Small growth rates of free groups

We introduce the notion of a Magnus marking for a finite generating set of a group and prove a certain expansion property.

Using this property, we determine the second and third smallest growth rates of the rank-$d$ free group for $d\ge 2$.

We also give a new lower bound for the smallest growth rate of the genus-$g$ surface group for $g\ge 2$, as well as a lower bound for the growth rate associated with a one-relator presentation.