Bounded displacement permutations on the integers

Abstract

It is shown that if each element of a group $G$ of permutations on $\mathbb{Z}$ displaces points by a bounded distance, then infinitely divisible elements of $G$ are torsion. One can replace the metric space $\mathbb{Z}$ with one which is sufficiently tree-like and having uniform bounds on the cardinalities of balls of a given radius. As a consequence we give a positive solution to a problem of N. M. Suchkov in the Kourovka Notebook.

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