Groups of given intermediate word growth

TL;DR

This paper constructs finitely generated groups with intermediate growth rates, covering a broad class of functions and introducing self-similar branched groups with specific growth exponents.

Contribution

It demonstrates the existence of finitely generated groups with growth functions matching a wide range of intermediate growth rates, including self-similar branched groups.

Findings

Existence of groups with growth ~f for functions satisfying specific inequalities.

Construction of self-similar branched groups with growth ~exp(R^).

Coverage of all functions growing faster than a certain exponential rate.

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(\eta R) for all R large enough and \eta\approx2.4675 the positive root of X^{3}-X^{2}-2X-4. This covers all functions that grow uniformly faster than \exp(R^{\log2/\log\eta}). We also give a family of self-similar branched groups of growth ~\exp(R^\alpha) for a dense set of \alpha\in(\log2/\log\eta,1).