## Homomorphisms from the harmonic archipelago group to finite groups

This post is a brief application of a result discussed in the last post about the existence of odd ways to map the fundamental group of the Hawaiian earring $\mathbb{H}$ onto an arbitrary finite group $G$:

Theorem 1: Let $G$ be any non-trivial finite group and $\ell_n$ be a loop going once around the n-th circle of the Hawaiian earring in the clockwise direction. There are uncountably many surjective group homomorphisms $\pi_1(\mathbb{H})\to G$ mapping $g_n=[\ell_n]$ to the identity element for every $n\geq 1$.

Since the kernel of each of these homomorphisms $\pi_1(\mathbb{H})\to G$ contains the infinite free group $F_{\infty}=F(g_1,g_2,...)$ generated by the classes $[\ell_n]$, there is clearly a connection to the harmonic archipelago $\mathbb{HA}$.

Harmonic Archipelago

Let’s fix a non-trivial finite group $G$ and show that Theorem 1 also holds for the Harmonic archipelago.

Recall that the canonical inclusion $\mathbb{H}\to\mathbb{HA}$ induces a surjective homomorphism $\phi:\pi_1(\mathbb{H})\to\pi_1(\mathbb{HA})$ (see Corollary 2 of this post). Moreover, $\ker\phi$ is the conjugate closure of the free group generated by the elements $g_{n}g_{n+1}^{-1}$, $n\geq 1$.

Of course, we have $g_{n}g_{n+1}^{-1}\in F_{\infty}$. So for every surjective homomorphism $f:\pi_1(\mathbb{H})\to G$ satisfying $f(F_{\infty})=1$, the inclusion $\ker\phi\subseteq\ker f$ holds and we get a unique surjective homomorphism $\overline{f}:\pi_1(\mathbb{HA})\to G$ such that $f=\overline{f}\circ\phi$.

Altogether, $\phi:\pi_1(\mathbb{H})\to\pi_1(\mathbb{HA})$ induces an injection

$\zeta=Hom(\phi,G):Hom(\pi_1(\mathbb{HA}),G)\to Hom(\pi_1(\mathbb{H}),G)$

which is pre-composition by $\phi$ and we have $\zeta(\overline{f})=f$ when $f:\pi_1(\mathbb{H})\to G$ is one of the (uncountably many) surjective homomorphisms guaranteed to exist by Theorem 1. This is enough to serve as the proof of the main theorem of this post.

Theorem 2: Let $G$ be any non-trivial finite group and $\ell_n$ be a loop going once around the n-th circle of $\mathbb{H}$ viewed as a subspace of $\mathbb{HA}$. There are uncountably many surjective group homomorphisms $\pi_1(\mathbb{HA})\to G$ mapping $[\ell_n]$ to the identity element for every $n\geq 1$.

Certainly then, we have uncountably many (overall) homomorphisms from $\pi_1(\mathbb{HA})$ to $G$.

Corollary 3: For any non-trivial finite group $G$, the set of group homomorphisms $Hom(\pi_1(\mathbb{HA}),G)$ is uncountable.

Theorem 2 and Corollary 3 are in stark contrast to the fact that the only homomorphism $\pi_1(\mathbb{HA})\to\mathbb{Z}$ to the additive group of integers is the trivial homomorphism (See Theorem 1 of The harmonic archipelago group is not free).

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