Groups with near exponential residual finiteness growth

TL;DR

This paper investigates the residual finiteness growth in groups acting on rooted trees, demonstrating near exponential growth in several important classes of branch groups, with some exhibiting super-exponential growth.

Contribution

It develops tools to identify near exponential residual finiteness growth in groups acting on rooted trees, including key examples like the Grigorchuk and Gupta-Sidki groups.

Findings

Near exponential residual finiteness growth in certain branch groups

Super-exponential residual finiteness growth in Gupta-Sidki p-groups for p≥5

Tools for recognizing residual finiteness growth in groups acting on rooted trees

Abstract

A function $\mathbb{N} \to \mathbb{N}$ is near exponential if it is bounded above and below by functions of the form $2^{n^c}$ for some $c > 0$. In this article we develop tools to recognize the near exponential residual finiteness growth in groups acting on rooted trees. In particular, we show the near exponential residual finiteness growth for certain branch groups, including the first Grigorchuk group, the family of Gupta-Sidki groups and their variations, and Fabrykowski-Gupta groups. We also show that the family of Gupta-Sidki p-groups, for $p\geq 5$, have super-exponential residual finiteness growths.