Non-uniform exponential growth and the decay of growth rates in growing dimensions

Abstract

We provide the first example of a finitely presented, and the first example of a simple, group of non-uniform exponential growth. The example is given by Thompson's group $V$. Our methods also show that the infimal exponential growth rates of $\mathrm{Aut}(F_{2^{n+2}})$ and of ${\mathrm{EL}}_{2^{n+2}}(R)$, for every finitely generated ring $R$, tend to $1$. As an application, we obtain the first example of an acylindrically hyperbolic group, and the first example of a Kazhdan group, of non-uniform exponential growth.

33 pages. This version merges and supersedes arXiv:2605.30163v1 and arXiv:2606.15945v1