Archipelago groups

The classical archipelago is a non-contractible subset of $\mathbb{R}^3$ which is homeomorphic to a disk except at one non-manifold point. Its fundamental group, $\mathcal{A}$, is the quotient of the topologist's product of $\mathbb Z$, the fundamental group of the shrinking wedge of countably many copies of the circle (the Hawaiian earring), modulo the corresponding free product. We show $\mathcal{A}$ is locally free, not indicable, and has the rationals both as a subgroup and a quotient group. Replacing $\mathbb Z$ with arbitrary groups yields the notion of archipelago groups. Surprisingly, every archipelago of countable groups is isomorphic to either $\mathcal{A}(\mathbb Z)$ or $\mathcal{A}(\mathbb Z_2)$, the cases where the archipelago is built from circles or projective planes respectively. We conjecture that these two groups are isomorphic and prove that for large enough cardinalities of $G_i$, $\mathcal{A}(G_i)$ is not isomorphic to either.

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