#### Vol. 304, No. 1, 2020

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Commensurability growth of branch groups

### Khalid Bou-Rabee, Rachel Skipper and Daniel Studenmund

Vol. 304 (2020), No. 1, 43–54
##### 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
##### Authors
 Khalid Bou-Rabee Department of Mathematics The City College of New York New York, NY United States Rachel Skipper Department of Mathematics Ohio State University Columbus, OH United States Daniel Studenmund Department of Mathematics University of Notre Dame Notre Dame, IN United States