In this post I’ll present some results and ideas from the paper The number of homomorphisms from the Hawaiian earring group, 2019. First I’ll give some motivating background and historical remarks, and then state the main result and concepts used in its proof.
The question “How many?” is an old one, and the ideas advanced by Cantor gave this question meaning in the infinite setting. Where previously an answer of “Infinitely many” was as descriptive as necessary, one can now ask the followup question “Which infinite?” When the answer is “as many as is naively allowable” then you get a sense of abundance. For example one might wonder how many different ultrafilters exist on an infinite set . There are collections of subsets of . Pospisil showed that there are ultrafilters on (see Remark on bicompact spaces, 1937) and since we have the greatest number imaginable.
When the question is “How many group homomorphisms are there from to ?”, a very naive upper bound is the total number of functions from to , namely . For example there are group homomorphisms from to and this is the same as the number of functions from to . One can see this by picking a basis for as a vector space over and considering all possible maps from basis elements to .
When considering other groups the answer can become more subtle. Recall that the Hawaiian earring is the space where is the circle of radius centered at the point . Its fundamental group, which we denote , is uncountable and not free; in fact any homomorphism to a free group will have finitely generated image. For each the group has a homomorphic retraction to a free subgroup of rank which is induced by the topological retraction map which takes all points which are not in the outermost circles of to the point .
A group is n-slender if for every homomorphism there exists some for which (see posts Automatic continuity parts 2 and 3). Right-angled Artin groups, Baumslag-Solitar groups, and torsion-free word hyperbolic groups are examples of n-slender groups, and groups containing torsion or a subgroup isomorphic to are not n-slender. A rough intuitive way of understanding an n-slender group is that homomorphisms from to can only be superficial- they cannot involve the infinitely deep parts of .
Curiously there exist models of Zermelo-Fraenkel set theory plus the axiom of dependent choices in which if is not n-slender we have (see Deeply concatenable subgroups might never be free, with Shelah). In particular, any homomorphism which witnesses that a group of cardinality less than is not n-slender is somehow “nonconstructive.” From this one suspects that such a homomorphism probably relies on performing many choices and is therefore not unique in witnessing n-slenderness (compare the construction of a non-principal ultrafilter). It turns out that there are as many homomorphisms from to a small cardinality group which is not n-slender:
Theorem If is a group with then
The requirement that is essential here, since is not n-slender (consider the identity map), , and (this latter equality follows from Corollary 2.11 of Eda’s Free -products and fundamental groups of subspaces of the plane, 1998). It was previously known that if is a nontrivial finite group then the number of surjections from to is (see Conner and Spencer, Anomalous behavior of the Hawaiian earring group, 2005). From more recent results one sees more generally that if is a compact Hausdorff topological group there exist many homomorphisms from to (using theorems from Tlas, Big free groups are almost free, 2015 and Zastrow, The non-abelian Specker group is free, 2000).
It is straightforward to show that if is n-slender then . The challenge lies in producing many homomorphisms from the existence of just one nonconstructive homomorphism. The trick for this lies in using the Harmonic archipelago (this space appeared in Bogley, Sieradski, Universal path spaces). This space can be imagined as taking the disc and from each subdisc you form a hill of height . This gives a space whose fundamental group is uncountable and onto which can be naturally surjected.
First, one shows that if and is not n-slender then there exists a nontrivial homomorphism (this fact was told to me by Greg Conner). Next one shows that there exist -many homomorphisms for which differs for distinct . This is the most difficult part of the argument and involves manipulation of infinitary words. Once one has shown this the theorem follows.