SSGP topologies on free groups of infinite rank

2019/02/03 by Dmitri Shakhmatov, Shakhmatov, Dmitri, Víctor Hugo Yañez +1
Mathematics · #22A25 #43A40 #43A60 #54H11 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Group Theory (math.GR) #Primary: 22A05 #Secondary: 20E05 #math.FA #math.GN #math.GR #msc:20E05 #msc:22A05 #msc:22A25 #msc:43A40 #msc:43A60 #msc:54H11

paper · pdf · doi:10.48550/arxiv.1902.00840

arxiv created 2019/02/03 · arxiv updated 2019/02/05

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 in G can be represented as a product g=g₁ g₂ ... g_k such that the cyclic group generated by each g_i 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.

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