Generating the inverse limit of free groups

Abstract

We study the relation between two uncountable groups with remarkable properties (cf. [15]): the topological free product of infinite cyclic groups G (the fundamental group of the Hawaiian Earring), and the inverse limit of finitely generated free groups Fˆ. The former has a canonical embedding as a proper subgroup of the latter and we examine when G, together with certain naturally defined normal subgroups of Fˆ generate the entire group Fˆ. We are interested in particular in normal subgroupsKerT(Fˆ)=⋂{Kerφ|φ∈hom⁡(Fˆ,T)}, where T is some finitely-presented n-slender group.Our main results state that if T is the infinite cyclic group or the free nilpotent class 2 group on 2 generators, then G and KerT(Fˆ) generate Fˆ. On the other hand, if T is the free nilpotent class 3 group or a Baumslag-Solitar group, then the product of subgroups G⋅KerTFˆ is a proper subgroup of Fˆ.In the last section, we provide an interesting geometric interpretation of the above results in terms of path-connectedness of certain fibrations arising as inverse limits of covering spaces over the Hawaiian earring space.