Branch groups, orbit growth, and subgroup growth types for pro-$p$ groups

TL;DR

This paper investigates the subgroup growth types of pro-$p$ branch groups, introduces orbit growth, and constructs extensions to demonstrate a wide range of possible subgroup growth behaviors, addressing a question by Lubotzky and Segal.

Contribution

It characterizes subgroup growth for certain pro-$p$ branch groups and introduces orbit growth to construct groups with diverse subgroup growth types.

Findings

Pro-$p$ branch groups have subgroup growth type $n^{\log n}$.

Extensions of Grigorchuk and Gupta-Sidki groups exhibit a broad spectrum of subgroup growth.

Almost all functions between $n^{(\log n)^2}$ and $e^n$ occur as subgroup growth types.

Abstract

We first show that a class of pro-$p$ branch groups including the Grigorchuk group and the Gupta-Sidki groups all have subgroup growth type $n^{\log n}$. We then introduce the notion of orbit growth and use it to construct extensions of the Grigorchuk group and the Gupta-Sidki groups. We compute the subgroup growth type of these extensions and deduce that all functions between $n^{(\log n)^2}$ and $e^n$ occur as the subgroup growth type of a pro-$p$ group, thus, giving almost complete answer to a question raised by Lubotzky and Segal.