Commensurability growth of branch groups

Fixing a subgroup $Γ$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Δ$ of $G$ with $[Γ: Γ\cap Δ][Δ: Γ\cap Δ] = n$. For pairs $Γ\leq A$, where $A$ is the automorphism group of a $p$-regular tree and $Γ$ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups $Γ$ (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 $n = p^k$.

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