Consider a group and a subgroup of . For an element , we denote the conjugation-by- map as . We say that commensurates if has a finite index in both and . The set of elements in that commensurate is called the commensurability group or the commensurator of in ; we denote it by . Define the th commensurator to be
The normalizer of in , which we denote by , acts on the left on the sets . We denote the size of the quotient by . The asymptotic behavior of the sequence is what we call the commensurator growth of the pair .
In the paper under review, the authors study the commensurator growth and it is motivated by the problem: Is there a function such that, for any lattice in a finitely generated group , its commensurator growth function satisfies for any ?
In this context, the authors prove the following theorem: let , then there exists a pair of groups such that for all . Moreover, several examples are given of commensurator growth for a locally compact topological group and a lattice inside it. In particular, for the pair , where is the three-dimensional Heisenberg group.