Commensurability growth of branch groups

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 $n = p^k$.