QUESTION AND HOMOMORPHISMS ON ARCHIPELAGO GROUPS (Set Theory and Infinity)

Abstract

The classical archipelago group is a quotient group of the fundamental group of the Hawaiian earring by the normal closure of the free group of countable rank, which is denoted by A.(Z). Since the fundamental group of the Hawaiian earring is expressed by the free σ-product XwZ, we obtain an archipelago group A(G) by replacing Z with G. In [1] the authors asserted that A(Z) and A(Z/kZ) are isomorphic for k≥3. We clarify a gap in their proof and show that there are surjective homomorphisms between A(Z/kZ)'s and A(Z) for k≥2. Finally we state our conjecture and some direction showing the conjecture.