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On commensurator growth. (English) Zbl 1260.22003

Consider a group G and a subgroup A of G. For an element gG, we denote the conjugation-by-g map as xx g =g -1 xg. We say that g commensurates A if AA 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(A,G). Define the nth commensurator to be

Comm n (A,G)={gComm(A,G)|A:AA g |=n}·

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

In the paper under review, the authors study the commensurator growth and it is motivated by the problem: Is there a function f: such that, for any lattice Γ in a finitely generated group G, its commensurator growth function satisfies c n (Γ,G)<f(n) for any n?

In this context, the authors prove the following theorem: let f:, then there exists a pair Γ<G of groups such that c n (Γ,G)f(n) for all 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 (H(),H()), where H is the three-dimensional Heisenberg group.

MSC:
22D05General properties and structure of locally compact groups
20E07Subgroup theorems; subgroup growth
20E08Groups acting on trees
20E34General structure theorems of groups
20F45Engel conditions on groups
20F18Nilpotent groups
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