SSGP topologies on free groups of infinite rank

Abstract

We prove that every free group G with infinitely many generators admits a Hausdorff group topology T with the following property: for every T-open neighbourhood U of the identity of G, each element g∈G can be represented as a product g=g1g2…gk, where k is a positive integer (depending on g) and the cyclic group generated by each gi is contained in U. In particular, G admits a Hausdorff group topology with the small subgroup generating property of Gould. This provides a positive answer to a question of Comfort and Gould in the case of free groups with infinitely many generators. The case of free groups with finitely many generators remains open.