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On commensurator growth. (English) Zbl 1260.22003

Consider a group $G$ and a subgroup $A$ of $G$. For an element $g\in G$, we denote the conjugation-by-$g$ map as $x↦{x}^{g}={g}^{-1}xg$. We say that $g$ commensurates $A$ if $A\cap {A}^{g}$ has a finite index in both $A$ and ${A}^{g}$. The set of elements in $G$ that commensurate $A$ is called the commensurability group or the commensurator of $A$ in $G$; we denote it by $Comm\left(A,G\right)$. Define the $n$th commensurator to be

${Comm}_{n}\left(A,G\right)=\left\{g\in \text{Comm}\left(A,G\right)\mid |A:A\cap {A}^{g}|=n\right\}·$

The normalizer of $A$ in $G$, which we denote by ${N}_{G}\left(A\right)$, acts on the left on the sets ${Comm}_{n}\left(A,G\right)$. We denote the size of the quotient ${Comm}_{n}\left(A,G\right)/{N}_{G}\left(A\right)$ by ${c}_{n}\left(G,A\right)$. The asymptotic behavior of the sequence ${c}_{n}\left(A,G\right)$ is what we call the commensurator growth of the pair $\left(A,G\right)$.

In the paper under review, the authors study the commensurator growth and it is motivated by the problem: Is there a function $f:ℕ\to ℕ$ such that, for any lattice ${\Gamma }$ in a finitely generated group $G$, its commensurator growth function satisfies ${c}_{n}\left({\Gamma },G\right) for any $n$?

In this context, the authors prove the following theorem: let $f:ℕ\to ℕ$, then there exists a pair ${\Gamma } of groups such that ${c}_{\le n}\left({\Gamma },G\right)\ge f\left(n\right)$ for all $n\in ℕ$. Moreover, several examples are given of commensurator growth for a locally compact topological group and a lattice inside it. In particular, for the pair $\left(H\left(ℤ\right),H\left(ℝ\right)\right)$, where $H$ is the three-dimensional Heisenberg group.

##### MSC:
 22D05 General properties and structure of locally compact groups 20E07 Subgroup theorems; subgroup growth 20E08 Groups acting on trees 20E34 General structure theorems of groups 20F45 Engel conditions on groups 20F18 Nilpotent groups
##### Keywords:
commensurator; commensurator growth; Heisenberg group
##### References:
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