Commensurability growth of branch groups

Khalid Bou-Rabee, Rachel Skipper and Daniel Studenmund

Vol. 304 (2020), No. 1, 43–54

DOI: 10.2140/pjm.2020.304.43

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 $\left[\Gamma :\Gamma \cap \Delta \right]\left[\Delta :\Gamma \cap \Delta \right]=n$. For pairs $\Gamma \le A$, where $A$ is the automorphism group of a $p$-regular rooted 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 rooted trees, this function is precisely ${\aleph }_0$ for any $n=p^k$.

Keywords

commensurators, branch groups, residually finite groups

Mathematical Subject Classification 2010

Primary: 20B07, 20E26

Secondary: 20K10

Milestones

Received: 30 August 2018

Revised: 20 July 2019

Accepted: 20 July 2019

Published: 18 January 2020