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

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.

finitegroup2Altogether, \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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This entry was posted in Cardinality, Finite groups, Fundamental group, harmonic archipelago and tagged , , , , , . Bookmark the permalink.

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