The ubiquity of free subgroups in certain inverse limits of groups
The main results of this paper are additions to an already large family of theorems establishing the abundance of free subgroups in structures of various kinds. In this paper we consider inverse limits of wreath products of arbitrary non-trivial groups and show that in a suitable sense ``almost all'' \(k\)-tuples in such inverse limits generate free groups of rank \(k\). Inverse limits of wreath products of groups are automorphism groups of suitably constructed rooted trees and can be shown to be complete metric spaces under a suitable metric. We prove the corresponding theorem for inverse limits \(W\) of wreath products of groups, namely: Theorem I. Let \(k\) be a positive integer and \(W\) an inverse limit of wreath products of non-trivial groups. Then the set of all \(k\)-tuples in \(W^k\) that do not freely generate free subgroups of rank \(k\) is a meagre subset of \(W^k\). If we restrict attention to inverse limits \(W\) of wreath products of finite groups, then we can consider them to be probability spaces, the probability being the normalized Haar measure. We also prove the analogue of Theorem I for the profinite case with respect to this natural measure on such spaces. More explicitly, we prove the following theorem. Theorem II. Let \(W\) be an inverse limit of wreath products of non-trivial finite groups, and let \(k\) be a positive integer. Then the set of \(k\)-tuples in \(W^k\) that do not generate free groups of rank \(k\) is a null set in the natural measure on \(W^k\).
scientific article; zbMATH DE number 763911