Grigorchuk's gap conjecture for finitely generated groups

Let $G=\langle X\rangle$ be a group generated by a finite set $X$, with growth function

$$\gamma_X(n)=|B_X(n)|,\qquad B_X(n)=\{g\in G:\ell_X(g)\leq n\}.$$

Write $f\prec g$ for the growth comparison used in the source: there is a constant $C>0$ such that $f(n)\leq Cg(Cn)$ for all $n>0$. A group is virtually nilpotent if it has a nilpotent subgroup of finite index. Grigorchuk’s gap conjecture. There is a constant $\beta>0$ such that, if

$$\gamma_X(n)\prec\exp(n^\beta),$$

then $G$ is virtually nilpotent.

This conjecture asks for a gap between polynomial growth and sufficiently slow subexponential growth. The paper also mentions the stronger version, which asserts that one may take $\beta=1/2$, and proves the conjecture for groups acting faithfully on bounded-degree rooted trees; conditional on Babai’s conjecture, it reduces the residually finite case to the simple case.