Generating the inverse limit of free groups

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 (F) over capF. 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 (F) over cap generate the entire group (F) over cap. We are interested in particular in normal sub-groups Ker(T) ((F) over cap) = boolean AND{Ker phi vertical bar phi is an element of hom((F) over cap, T)}, where T is some finitely-presented n-slender group. Our main results state that if Tis the infinite cyclic group or the free nilpotent class 2 group on 2 generators, then G and Ker(T)((F) over cap) generate (F) over cap. On the other hand, if T is the free nilpotent class 3 group or a Baumslag-Solitar group, then the product of subgroups G . Ker(T) (F) over cap is a proper subgroup of (F) over cap. 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. (C) 2021 Elsevier Inc. All rights reserved.