Groups of given intermediate word growth

Abstract

We show that there exists a finitely generated group of growth ~f for all functions f:\mathbb{R}\rightarrow\mathbb{R} satisfying f(2R) \leq f(R)^{2} \leq f(ηR) for all R large enough and η\approx2.4675 the positive root of X^{3}-X^{2}-2X-4. This covers all functions that grow uniformly faster than \exp(R^{\log2/\logη}). We also give a family of self-similar branched groups of growth ~\exp(R^α) for a dense set of α\in(\log2/\logη,1).