The fundamental groups of one-dimensional wild spaces and the Hawaiian earring (Q2781314)

Let \(I\) be an index set, and let \(\{G_i \mid i\in I\}\) be a set of groups. Roughly speaking, a \(\sigma\)-word in the \(G_i\) is a countable string \(\dots g_r g_{r+1} \dots\) (\(r \in \mathbb Z\)) with \(g_r \in G_i\), and for fixed \(i\), there are only finitely many \(r\) such that \(g_r \in G_i\). Then in a similar fashion that one constructs free products, one can construct the free \(\sigma\)-product \(\pmb \times_{i\in I}^{\sigma} G_i\) whose elements consist of equivalence classes of \(\sigma\)-words, and product is given by concatenation. In the case \(I\) is finite, the \(\sigma\)-product is the same as the usual free product. For \(i \in \mathbb Z\), let \(\mathbb Z_i\) denote a copy of the group \(\mathbb Z\). Then we say that a group is n-slender if and only if any homomorphism \(\pmb \times_{i \in \mathbb Z} \mathbb Z_i \to G\) depends on only finitely many coordinates. The class of n-slender groups includes \(\mathbb Z\) and is closed under free products and direct sums. On the other hand n-slender groups are torsion free, and the \(p\)-adic integers is not n-slender for any prime \(p\). The main result of this paper isNEWLINENEWLINENEWLINETheorem 1.1 Let \(X\) be a one-dimensional space which contains a copy \(C\) of the circle, let \(x_0 \in C\), and suppose \(X\) is not semi-locally simply connected at any point on \(C\). Then the fundamental group \(\pi_1(X,x_0)\) cannot be embedded in \(\pmb \times_{i\in I}^{\sigma}G_i\) for n-slender groups \(G_i\).NEWLINENEWLINENEWLINELet \(\mathbb H\) denote the Hawaiian earring, that is \(\bigcup_{n=1}^{\infty} \{(x,y) \mid (x+1/n)^2 + y^2 = 1/n^2\}\). \textit{K. Eda} [J. Algebra 148, No. 1, 243-263 (1992; Zbl 0779.20012)] has shown that the fundamental group of \(\mathbb H\) is isomorphic to \(\pmb \times_{i \in \mathbb Z} \mathbb Z_i\). It now follows from Theorem 1.1 that the fundamental groups of the Sierpinski gasket, Sierpinski curve and Menger curve are not embeddable in the fundamental group of \(\mathbb H\). This answers Question 3.5.1 of \textit{J. W. Cannon} and \textit{G. R. Conner} [preprint]. Since Cannon and Conner have proved that an embedding between one-dimensional Hausdorff spaces induces an embedding between their fundamental groups [\textit{J. W. Cannon} and \textit{G. R. Conner}, preprint, Corollary 3.3], we now see that the Sierpinski gasket, Sierpinski curve and Menger curve cannot be embedded in \(\mathbb H\).