This is a guest post by Sam Corson, who is a Heibronn Fellow at the University of Bristol.
This first post will provide background on the infinite word combinatorics which are used in the description of the fundamental group of each of the spaces in question. The Griffiths twin cone space first appeared in print in H. B. Griffith’s paper The fundamental group of two spaces with a common point, Quart. J. Math. Oxford 2 (1954), 175-190. The first appearance of the harmonic archipelago seems to be in the work of W. A. Bogley and A. J. Sieradski Weighted combinatorial group theory and wild metric complexes, Groups-Korea ’98 (Pusan), de Gruyter, Berlin, 2000, 53-80. For more background into these two spaces, you can consult some of Brazas’ old blog posts: harmonic archipelago and Griffiths twin cone. The conjecture that originated with James W. Cannon and Gregory R. Conner.
Recall that the earring space is the shrinking wedge of countably infinitely many circles. More formally if we let denote the circle centered at of radius . The subspace is given by (this post of Brazas gives some nice background). It is well-known that the fundamental group of a wedge of circles is a free group (with each circle corresponding to a free generator), and so one would expect that the the fundamental group will be like a free group. While this is true, it is emphasized that is not a free group. This is best illustrated by the curious fact that and cannot homomorphically surject onto a free group of infinite rank (for this latter result, see Theorem 1 of G. Higman, Unrestricted free products and topological varieties, J. London Math. Soc. 27 (1952), 73-81.)
Let be a countably infinite collection of symbols, which we will call letters, which is equipped with formal inverses. Usually the superscript is not written. A word is a finite-to-one function where the domain is a totally ordered set (finite-to-one means in this case that for each and the set is finite). It follows that the domain of a word must be countable (possibly finite or empty). As an example the infinite string
is a word; more formally it is the word given by (notice that each element of the alphabet is utilized at most once in the word). The infinite string
given by the rule is not a word since the letter is used infinitely often. Let denote the empty word, i.e. the word with empty domain. A word can have more exotic domain than : any finite-to-one function is a word. As a technical aside, we consider two words and to be equivalent, and write , provided there exists an order isomorphism such that for all . We form the concatenation of two words and , denoted , by declaring that has domain which is the disjoint union with the elements in being ordered below those in and having
Analogously, given a totally ordered set and collection of words indexed by we can form a function whose domain is the disjoint union , ordered in the natural way, and defined by where . This function we denote and it is a word provided it is finite-to-one.
A word has an inverse, which is denoted , given by letting be the set under the reverse order and . For example the inverse of the word
will be the word
Given and word we let be the finite word given by the restriction . Given words we write if for each the words and are equal as elements in the free group. For example, the word
has , and for we get
It is easy to see that .
The group is isomorphic to the collection of equivalence classes over . The binary operation is given by concatenation: and the class of the empty word plays the role of the group identity. Inverses in the group are predictably defined by .
Analogously to a free group, there are specific words with which we prefer to work. Given a word we say that is a subword of if there exist words (either or both of which may be empty) such that . Moreover is an initial (respectively terminal) subword provided (resp. ) in the above writing is empty. Finally a word is reduced if for every subword we have implies . Clearly every subword of a reduced word is itself reduced. The proof of the following result is far more difficult than that of the free group analogue:
Lemma. Every class contains a reduced word which is unique up to . Letting denote the reduced representative of the class of word we have for all words that . Moreover, given reduced words there exist words such that
(4) is reduced.
For further reading on (reduced) words see Section 1 of K. Eda, Free -products and noncommutatively slender groups, J. Algebra 148 (1992), 243-263.
The nice qualities of reduced words motivate one to consider the earring group as the set of reduced words with binary operation . We introduce two alphabets with formal inverses:
(with for “h”armonic archipelago); and
(with for “t”win cone).
Define words, concatenation, , reduced word, etc. just as before for each of these new alphabets and let and denote the respective sets of reduced words. These two sets are each groups under the binary operation and both are isomorphic to (the isomorphism with is given by the word mapping which extends and the isomorphism with is given by where ).
A word is pure if the first subscript in each of the letters appearing in is , and –pure is defined analogously. A word is –pure provided it is either -pure or -pure. For every subword of a -pure word is again -pure, and the only word which is both -pure and -pure is . Let denote the set of -pure words. The group is isomorphic to , where the notation denotes the smallest normal subgroup which includes the input. This isomorphism can be seen by two applications of van Kampen’s Theorem (see e.g. Section 4 in K. Eda, H. Fischer, Cotorsion-free groups from a topological viewpoint, Topology Appl. 214 (2016), 21-34.)
A word is –pure, where , provided all subscripts of letters appearing in are (i.e. is of form where ). A word is –pure provided it is -pure for some and we let denote the set of -pure words. The group is isomorphic to (see Theorem 5 of G. R. Conner, W. Hojka, M. Meilstrup, Archipelago groups, Proc. Amer. Math. Soc. 143 (2015), 4973-4988.)
Now the task of establishing the isomorphism is reduced to producing an isomorphism between and . This is not an easy task. It’s a nice exercise to check that any continuous function induces a trivial homomorphism (using the fact that is a Peano continuum and any continuous Hausdorff image of a Peano continuum is again a Peano continuum). While it is possible to give a continuous function so that is surjective, it is not possible to make such an injective as well. Thus, the natural (spacial) homomorphisms are ruled out. The fact that each element of is a (possibly infinitary) concatenation of -pure words and similarly each element of is a (possibly infinitary) concatenation of -pure words should be used in some way. A confounding issue is that and . We will continue in Part 2.