The gap conjecture

Statement

Beyond Gromov's theorem one can ask whether there exists a gap in the growth spectrum for finitely generated group just above polynomial growth, separating virtually nilpotent groups from others. Formally, this means that there would exist a function $f: \mathbb N \to \mathbb N$ such that a finitely generated group is virtually nilpotent if and only if its growth function is an $O(f(n))$. Such a theorem was obtained by Shalom and Tao, with an explicit function $n^{\log\log(n)^c}$ for some $c > 0$. All known groups with intermediate growth (i.e. both superpolynomial and subexponential) are essentially generalizations of Grigorchuk's group, and have faster growth functions; so all known groups have growth faster than $e^{n^{\alpha-o(1)}}$, with $\alpha = \log(2)/\log(2/\eta ) \approx 0.767$, where $\eta$ is the real root of the polynomial $x^3+x^2+x-2$.

It is conjectured that the true lower bound on growth rates of groups with intermediate growth is $e^{\sqrt n}$. This is known as the Gap conjecture.