Commensurability growth of branch groups

TL;DR

This paper investigates the commensurability growth function for subgroups of automorphism groups of p-regular trees, revealing diverse cardinalities and specific behaviors for well-known branch groups.

Contribution

It characterizes the possible cardinalities of the commensurability growth function and determines its exact value for several prominent branch groups.

Findings

The function can be finite, countable, or uncountable.

For many known branch groups, the function equals countably infinite for all p^k.

The behavior varies depending on the subgroup and the ambient automorphism group.

Abstract

Fixing a subgroup $\Gamma$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ of $G$ with $[\Gamma: \Gamma \cap \Delta][\Delta : \Gamma \cap \Delta] = n$. For pairs $\Gamma \leq A$, where $A$ is the automorphism group of a $p$-regular tree and $\Gamma$ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups $\Gamma$ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.) acting on $p$-regular trees, this function is precisely $\aleph_0$ for any $n = p^k$.