Groups where free subgroups are abundant

TL;DR

This paper investigates the property of topological groups where free subgroups are dense, introduces the concept of almost k-free groups, and provides examples and characterizations of such groups, extending previous results.

Contribution

It introduces the notion of almost k-free groups, constructs examples, and generalizes prior work on free subgroups in topological groups.

Findings

Non-discrete Hausdorff groups with dense free subgroups are almost k-free.

Permutation groups of infinite sets are almost 2^|X|-free.

Infinite topological groups are almost aleph_0-free iff they are almost n-free for all n.

Abstract

Given an infinite topological group G and a cardinal k>0, we say that G is almost k-free if the set of k-tuples in G^k which freely generate free subgroups of G is dense in G^k. In this note we examine groups having this property and construct examples. For instance, we show that if G is a non-discrete Hausdorff topological group that contains a dense free subgroup of rank k>0, then G is almost k-free. A consequence of this is that for any infinite set X, the group of all permutations of X is almost 2^|X|-free. We also show that an infinite topological group is almost aleph_0-free if and only if it is almost n-free for each positive integer n. This generalizes the work of Dixon and Gartside-Knight.