## The Griffiths twin cone and the harmonic archipelago have isomorphic fundamental group (Part 3)

We saw in the previous post that the problem of producing our isomorphism is solved provided we can produce a sufficiently large coherent collection of coi triples. But how is this to be accomplished? For example, given a (perhaps quite complicated) word $W \in \textbf{Red}_T$, is there a way to find some $U \in \textbf{Red}_H$ and coi $\iota$ from $W$ to $U$ so that the one-element collection $\{\text{coi}(W, \iota, U)\}$ is coherent? More challengingly, if we have already defined a coherent collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ of coi triples and we are given a word $W \in \textbf{Red}_T$ then can we find $\iota$ and $U \in \textbf{Red}_H$ so that the slightly larger collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is again coherent? And even if we can surmount this challenge for a reasonable coherent collection, might we still fail to produce a sufficiently large coherent collection on account of the fact that

$\beth_T(\text{P-fine}(\{U_x\}_{x \in X})) = \textbf{Red}_T/\langle\langle \textbf{Pure}_T \rangle\rangle$

but

$\beth_H(\text{P-fine}(\{W_x\}_{x \in X})) \neq \textbf{Red}_H/\langle\langle\textbf{Pure}_H\rangle\rangle$.

In other words, we may have exhausted the codomain but have failed to fully extend the homomorphism to have the appropriate domain. The reverse problem could also occur: we could exhaust the codomain before producing the isomorphism.

The last two potential problems are solved by alternately considering the elements of $\textbf{Red}_H$ and $\textbf{Red}_T$, ensuring that no $[[\cdot]]$-classes of words are left out of the homomorphism by a transfinite induction. The addition of “just one more coi” can require a great deal of technical care, and we will attempt to give the big ideas behind the ability to do so. We let $\|W\| = \frac{1}{n +1}$ where $n$ is the smallest subscript on a letter in $W \in \textbf{Red}_H$ (and $\|E\| = 0$) and similarly $\|U\| = \frac{1}{n + 1}$ where $n$ is the smallest second subscript of a letter in the word $U \in \textbf{Red}_T$.

To begin our collection of coi we notice that $\{\text{coi}(W, \iota_W, E)\}_{W \in \textbf{Pure}_H}$ is coherent (each $\iota_W$ is obviously the empty function). So far our collection is countable (since $|\textbf{Pure}_H| = \aleph_0$) and more particularly of cardinality less than $2^{\aleph_0}$. Next one can prove the following (we’ll number lemmas within this post).

Lemma 1. Suppose that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent and that $\epsilon > 0$.

(1) If $W \in \text{P-fine}(\{W_x\}_{x \in X})$ then we can find $U \in \textbf{Red}_T$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)$ is coherent, and $\|U\| \leq \epsilon$, and $U \not\equiv E$ provided $W \not\equiv E$.

(2) If $U \in \text{P-fine}(\{U_x\}_{x \in X})$ then we can find $W \in \textbf{Red}_H$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)$ is coherent, and $\|W\| \leq \epsilon$, and $W \not\equiv E$ provided $U \not\equiv E$.

The proof of this not-very-surprising lemma uses the fact that changing finitely many pure p-chunks of a word does not change the $[[\cdot]]$ equivalence class. Next we tackle infinitary concatenations of order type $\mathbb{N}$ (and we will need to use the crucial fact that the coi collection is not very large).

Lemma 2. Suppose that $\{ \text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent, $\textbf{Pure}_H \subseteq \{W_x\}_{x \in X}$, and $|X| < 2^{\aleph_0}$.

(1) If $W \in \textbf{Red}_H \setminus \text{P-fine}(\{W_x\}_{x \in X})$ and we can write $\text{p-index}(W) \equiv \prod_{n \in \mathbb{N}} I_n$ with each $I_n \neq \emptyset$ and $W\upharpoonright_p I_n \in \text{P-fine}(\{W_x\}_{x \in X})$, then we can find $U \in \textbf{Red}_{T}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

(2) If $U \in \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x\in X})$ and we can write $\text{p-index}(U) \equiv \prod_{n \in \mathbb{N}} I_n$ with each $I_n \neq \emptyset$ and $U \upharpoonright_p I_n \in \text{P-fine}(\{U_x\}_{x \in X})$, then we can find $W \in \textbf{Red}_H$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

To prove part (1) we inductively use Lemma 1 (1) to produce a coherent collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W\upharpoonright_p I_n, \iota_n, U_n)\}_{n \in \mathbb{N}}$ so that $U_n \neq E$ and $\|U_{n+1}\| \leq \frac{\|U_n\|}{2}$. Now an obvious candidate for $U$ would be $\prod_{n \in \mathbb{N}}U_n$, and this infinitary concatenation is indeed a word by the requirement $\|U_{n+1}\| \leq \frac{\|U_n\|}{2}$, but it may not be reduced. Therefore we instead will introduce a sequence of words $\{V_n\}_{n \in \omega} \subseteq \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x\in X} \cup \{U_n\}_{n\in \mathbb{N}})$ with $\|V_n\| = \|U_n\|$ and $|\text{p-index}(V_n)| \in \{1, 2\}$ and so that each concatenation $U_nV_nU_{n + 1}$ is reduced. The ability to make such a selection is guaranteed be the fact that the number of pure elements in $\text{P-fine}(\{U_x\}_{x\in X} \cup \{U_n\}_{n\in \mathbb{N}})$ is at most $(|X| + |\mathbb{N}|) \cdot \aleph_0 < 2^{\aleph_0}$. The fact that

$U \equiv U_0V_0U_1V_1\cdots$

is reduced uses the fact that each subword $U_nV_nU_{n + 1}$ was reduced (and we allowed $\text{p-index}(V_n)$ to have cardinality either $1$ or $2$ depending on how the word $U_n$ ends and how the word $U_{n + 1}$ begins). The function $\iota$ will be given in the obvious way: $\iota = \bigcup_{n \in \mathbb{N}} \iota_n$ and the tedious check that

$\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W\upharpoonright_p I_n, \iota_n, U_n)\}_{n \in \omega} \cup \{\text{coi}(W, \iota, U)\}$

is coherent (and therefore so is $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$) uses the fact that $\{V_n\}_{n \in \omega} \subseteq \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x\in X} \cup \{U_n\}_{n\in \mathbb{N}})$.

The proof for part (2) is somewhat similar: one inductively extends to a larger coherent collection

$\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W_n, \iota_n, U \upharpoonright_p I_n )\}_{n \in \omega}$

using Lemma 1 (2), but “buffer” words $V_n \in \textbf{Red}_H$ are selected during the induction to be of form $V_n \equiv h_{k_n}^{m_n}$. The sequences $\{k_n\}_{n \in \mathbb{N}}$ and $\{m_n\}_{n \in \mathbb{N}}$ are selected so that for each $n$ we have

$W_nV_nW_{n + 1}V_{n + 1}\cdots \notin \text{P-fine}(\{W_x\}_{x\in X} \cup \{W_n\}_{n \in \mathbb{N}})$

(this selection makes use of the fact that $|X| < 2^{\aleph_0}$).

Another difficult situation arises with concatenations which are of order type $\mathbb{Q}$.

Lemma 3. Suppose that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent, $\textbf{Pure}_H \subseteq \{W_x\}_{x \in X}$, and $|X| < 2^{\aleph_0}$.

(1) If $W \in \textbf{Red}_H \setminus \text{P-fine}(\{W_x\}_{x \in X})$ and we can write $\text{p-index}(W) \equiv \prod_{q \in \mathbb{Q}} I_q$ with each $I_q \neq \emptyset$ and $W \upharpoonright_p I_n \in \text{P-fine}(\{W_x\}_{x \in X})$ and $I_q$ is a maximal such interval, then we can find $U \in \textbf{Red}_{T}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

(2) If $U \in \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x \in X})$ and we can write $\text{p-index}(U) \equiv \prod_{q \in \mathbb{Q}} I_q$ with each $I_q \neq \emptyset$ and $U \upharpoonright_p I_n \in \text{P-fine}(\{U_x\}_{x \in X})$ and $I_q$ is a maximal such interval, then we can find $W \in \textbf{Red}_{H}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

For (1) we make a list $\{W_n\}_{n\in \mathbb{N}}$ so that for each $q\in \mathbb{Q}$ exactly one of $W \upharpoonright_p I_q$ or $(W \upharpoonright_p I_q)^{-1}$ appears in the enumeration. As in Lemma 2 we produce a coherent collection

$\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W_n, \iota_n, U_n )\}_{n \in \omega}$

by inductively using Lemma 1 and the sequence $\{U_n\}_{n \in \mathbb{N}}$ is again selected to satisfy nice properties; for example the values $\|U_n\|$ shrink to $0$ quite rapidly. Now we select two buffer words $V_{n, a}, V_{n, b}$, this time for both the front and tail of the word $U_n$, so that $V_{n, a}U_nV_{n, b}$ is reduced and some other technical properties hold. Now define the word $U \equiv \prod_{q\in \mathbb{Q}} (V_{f(q), a}U_f(q)V_{f(q), b})^{\delta_q}$ where $W \upharpoonright_p I_q \in \{W_{f(q)}^{\pm 1}\}$ and $\delta_q \in \{\pm 1\}$ with $\delta_q = 1$ if and only if $W \upharpoonright_p I_q \equiv W_{f(q)}$. From how cleverly the buffer words were selected, one argues that $U$ is reduced, and a coi $\iota$ is produced from the collection $\{\iota_n\}_{n \in \mathbb{N}}$ in the natural way. Part (2) requires similar modifications as those used in Lemma 2 (2). In both (1) and (2) the ability to select suitably nice buffer words makes essential use of the fact that $|X| < 2^{\aleph_0}$.

Lemma 4. Suppose that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ is coherent, $\textbf{Pure}_H \subseteq \{W_x\}_{x \in X}$, and $|X| < 2^{\aleph_0}$.

(1) If $W \in \textbf{Red}_H \setminus \text{P-fine(}\{W_x\}_{x \in X})$ then there exists $U \in \textbf{Red}_T$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

(2) If $U \in \textbf{Red}_T \setminus \text{P-fine}(\{U_x\}_{x \in X})$ then there exists $W \in \textbf{Red}_{H}$ and $\iota$ so that $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

The proof of part (2) is essentially that of part (1), with obvious modifications. For (1) we ask whether there exists a sequence of intervals $\{I_n\}_{n \in \mathbb{N}}$ in $\text{p-index}(W)$ where all $I_n$ have the same minimum or all have the same maximum, $I_n$ is properly included into $I_{n + 1}$, $W \upharpoonright_p I_n \in \text{P-fine}(\{W_x\}_{x \in X})$ for all $n$, and $W \upharpoonright_p \bigcup_{n\in \mathbb{N}} I_n \notin \text{P-fine}(\{W_x\}_{x \in X})$. If such an interval does not exist then we proceed to the next paragraph. If it does exist, then we extend the coi collection so as to include $W \upharpoonright_p \bigcup_{n\in \mathbb{N}} I_n$ using Lemma 2 (1) (applied to $(W \upharpoonright_p \bigcup_{n\in \mathbb{N}} I_n)^{-1}$ in case all the $I_n$ have a common maximum) and we once again ask whether such a sequence exists for the new collection. We do this over and over again, taking unions of the previously defined coherent collections at limit ordinals. Using certain parameters to keep track of how many times this process iterates, we deduce that it can only be executed countably many times. Thus we move on to the next step.

If $W$ is in $\text{P-fine}(\{W_y\}_{y\in Y})$, where $\{\text{coi}(W_y, \iota_y, U_y)\}_{y \in Y}$ is the slightly enlarged coi collection, then we produce $U$ and $\iota$ using Lemma 1 (1). Else we can write $\text{p-index}(W) \equiv \prod_{\lambda \in \Lambda} I_{\lambda}$ where $\Lambda$ is infinite dense-in-itself and each interval $I_{\lambda}$ is nonempty and maximal such that $W \upharpoonright_p I_{\lambda} \in \text{P-fine}(\{W_y\}_{y\in Y})$. The set $\Lambda$ may have a maximum and/or minimum, so we let $\Lambda' \subseteq \Lambda$ be the subset excluding such elements. Then $\Lambda' \equiv \mathbb{Q}$ and we use Lemma 4 (1) to extend to a collection, say, indexed by $Y'$, so that $W\upharpoonright_p \bigcup_{\lambda \in \Lambda'} I_{\lambda} \in \text{P-fine}(\{W_y\}_{y \in Y'})$ and by applying Lemma 1 (1) perhaps once or twice (in case we have a maximum and/or minimum in $\Lambda$) we then obtain the $U$ and $\iota$ so that the collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X} \cup \{\text{coi}(W, \iota, U)\}$ is coherent.

Now that we are armed with Lemma 4 we can define a suitable collection by induction over $2^{\aleph_0}$. Let $\prec_H$ (respectively $\prec_T$) be a well-ordering of $\textbf{Red}_H$ (resp. $\textbf{Red}_T$) such that each element has fewer than $2^{\aleph_0}$ predecessors. We already have $\{\text{coi}(W_n, \iota_n, E)\}_{m \in \mathbb{N}}$ in our collection, where $\textbf{Pure}_H = \{W_m\}_{m \in \mathbb{N}}$ is an enumeration. Recall that each ordinal $\alpha$ can be expressed uniquely as $\alpha = \gamma + n$ where $\gamma$ is either zero or a limit ordinal and $n \in \mathbb{N}$; in particular each ordinal can be considered either even or odd depending on the number $n \in \mathbb{N}$.

Suppose that we have already defined a coherent collection $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < \zeta}$ for all $\zeta < \alpha$ where $\mathbb{N} \leq \alpha$ is an ordinal below $2^{\aleph_0}$. Then the collection $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < \alpha}$ is coherent (this is easy to check). If $\alpha$ is even then we select $W \in \textbf{Red}_H$ such that $[[W]] \notin \beth_H(\text{Pfine}(\{W_{\beta}\}_{\beta < \alpha}))$ (such a $W$ exists using a cardinality argument) which is minimal under $\prec_H$ and by Lemma 4 (1) we choose suitable $U$ and $\iota$ to coherently extend and write $W_{\alpha} \equiv W$, $\iota_{\alpha} = \iota$, and $U_{\alpha} \equiv U$. In case $\alpha$ is odd we instead select $U \in \textbf{Red}_T$ with $[[U]] \notin \beth_T(\text{P-fine}(\{U_{\beta}\}_{\beta < \alpha}))$ which is minimal under $\prec_T$, use Lemma 4 (2) and extend accordingly. Thus we obtain a larger coherent collection $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < \alpha + 1}$.

Perform the process on all $\alpha < 2^{\aleph_0}$ and it is clear that $\{\text{coi}(W_{\beta}, \iota_{\beta}, U_{\beta})\}_{\beta < 2^{\aleph_0}}$ is coherent and

$\beth_H(\text{P-fine}(\{W_{\beta}\}_{\beta < 2^{\aleph_0}})) = \textbf{Red}_H/\langle\langle\textbf{Pure}_H\rangle\rangle$

and similarly

$\beth_T(\text{P-fine}(\{U_{\beta}\}_{\beta < 2^{\aleph_0}})) = \textbf{Red}_T/\langle\langle\textbf{Pure}_T\rangle\rangle$.

The argument is finished.

## The Griffiths twin cone and the harmonic archipelago have isomorphic fundamental group (Part 2)

This is Part 2 of a guest post by Sam Corson, who is a Heibronn Fellow at the University of Bristol. It will be helpful to read Part 1 first.

We will furthermore overload the notation used for word concatenation and apply it to totally ordered sets. For example, we will write $\Lambda_0 \equiv \Lambda_1$ for totally ordered sets $\Lambda_0$ and $\Lambda_1$ provided there exists an order isomorphism between them. The concatenation of totally ordered sets $\Lambda_0$ and $\Lambda_1$ is denoted $\Lambda_0\Lambda_1$ and is the disjoint union $\Lambda_0 \sqcup \Lambda_1$ under the obvious order. If $\{\Lambda_{\lambda}\}_{\lambda \in \Lambda}$ is a collection of totally ordered sets, indexed by a totally ordered set $\Lambda$, then the concatenation $\prod_{\lambda \in \Lambda} \Lambda_{\lambda}$ is the totally ordered set which is the disjoint union $\bigsqcup_{\lambda \in \Lambda} \Lambda_{\lambda}$ under the natural order.

Given a word $W \in \textbf{Red}_H$ and $t \in \overline{W}$ there exists a maximal interval $I \subseteq \overline{W}$ such that $t \in I$ and $W \upharpoonright I$ is $H$-pure. We can therefore write $\overline{W} \equiv \prod_{\lambda \in \Lambda} I_{\lambda}$ where each $I_{\lambda}$ is a maximal nonempty interval in $\overline{W}$ for which $W \upharpoonright I_{\lambda}$ is $H$-pure, and the totally ordered set $\Lambda$ is unique up to $\equiv$. This gives rise to a decomposition of the word $W$ as $W \equiv \prod_{\lambda \in \Lambda} W_{\lambda}$ where $W_{\lambda} \equiv W \upharpoonright I_{\lambda}$. This decomposition we call the p-decomposition and write $W \equiv_p \prod_{\lambda \in \Lambda} W_{\lambda}$ to express that the p-decomposition of $W$ is the concatenation $\prod_{\lambda \in \Lambda} W_{\lambda}$. We let $\text{p-index}(W)$ denote the totally ordered set $\Lambda$, which is well-defined up to $\equiv$. Of course, $\text{p-index}(E) \equiv \emptyset$. As an example, one can consider the word

$W\equiv a_0^1a_1^{-2}a_2^{3}a_3^{-4}a_4^{-5}\cdots$

where one has $\text{p-index}(W) \equiv \mathbb{N}$ and $W \equiv_p \prod_{n \in \mathbb{N}} W_n$ where $W_n \equiv a_n^{(n+1)(-1)^n}$.

A word $W'$ is a p-chunk of the word $W \equiv_p \prod_{\lambda \in \text{p-index}(W)} W_{\lambda}$ provided there exists some interval $J \subseteq \text{p-index}(W)$ such that $W' \equiv \prod_{\lambda \in J} W_{\lambda}$ (we may indeed write $W' \equiv_p \prod_{\lambda \in J} W_{\lambda}$). Thus a p-chunk of a word is a subword which respects the p-decomposition. Given an interval $J \subseteq \text{p-index}(W)$ we write $W\upharpoonright_p J$ for the p-chunk $\prod_{\lambda \in J} W_{\lambda}$. An $H$-pure p-chunk of a word $W \equiv_p \prod_{\lambda \in \text{p-index}(W)}W_{\lambda}$ will clearly either be $E$ or will be one of the $W_{\lambda}$. Given $W \in \textbf{Red}_H$ we will let $\text{p-chunk}(W)$ denote the set of all p-chunks of $W$. Note that this set might be uncountable (if $\text{p-index}(W) \equiv \mathbb{Q}$ then consider the p-chunks associated with the Dedekind cuts). Given a subset $\{W_x\}_{x \in X} \subseteq \textbf{Red}_H$ we let $\text{P-fine}(\{W_x\}_{x \in X})$ denote the generated subgroup $\langle \bigcup_{x \in X} \text{p-chunk}(W_x) \rangle \leq \textbf{Red}_H$. One can prove that this subgroup is closed under taking p-chunks of elements.

For $U \in \textbf{Red}_T$ there similarly exists a decomposition of $\overline{U}$ into maximal nonempty intervals $\overline{U} \equiv \prod_{\lambda \in \Lambda} I_{\lambda}$ where $U \upharpoonright I_{\lambda}$ is $T$-pure. Thus we obtain a decomposition, which we again call the p-decomposition and use the same notation $\equiv_p$ and again write $\text{p-index}(U)$ to identify the p-index.

If $W \in \textbf{Red}_H$ with $W \equiv_p \prod_{\lambda \in \text{p-index}(W)}W_{\lambda}$ and $F \subseteq \text{p-index}(W)$ is finite then we have $[[W]] = [[\textbf{Red}(\prod_{\lambda \in \text{p-index}(W) \setminus F}W_{\lambda})]]$, and similarly for a word $U \in \textbf{Red}_T$. This fact that the $[[\cdot]]$ class is preserved under deleting finitely many elements of the p-index and then reducing provides the motivation for the essential idea in constructing the isomorphism. We recount the idea now.

Given a totally ordered set $\Lambda$ we will say that a subset $\Lambda' \subseteq \Lambda$ is close in $\Lambda$ if for every infinite interval $I \subseteq \Lambda$ we have $I \cap \Lambda' \neq \emptyset$. For example, if $\Lambda$ is finite then every subset of $\Lambda$, including $\emptyset$, is close in $\Lambda$. If $\Lambda \equiv \mathbb{N}$ then any infinite subset $\Lambda' \subseteq \Lambda$ is close in $\Lambda$. If $\Lambda \equiv \mathbb{Q}$ then every dense subset of $\Lambda$ will be close in $\Lambda$. If $\Lambda_0$ and $\Lambda_1$ are totally ordered sets and $\Lambda_0' \subseteq \Lambda_0$ and $\Lambda_1' \subseteq \Lambda_1$ are each close then we call an order isomorphism $\iota: \Lambda_0' \rightarrow \Lambda_1'$ a close order isomorphism (abbreviated coi) from $\Lambda_0$ to $\Lambda_1$.

A close order isomorphism $\iota$ from $\Lambda_0$ to $\Lambda_1$ defines a correspondence (not necessarily one-to-one) between the intervals of $\Lambda_0$ and those in $\Lambda_1$: given interval $I \subseteq \Lambda_0$ we obtain interval $\varpropto(I, \iota) := \bigcup_{\lambda < \lambda'; \lambda, \lambda' \in \iota(I)} [\lambda, \lambda'] \subseteq \Lambda_1$. For an interval $I \subseteq \Lambda_1$ we define $\varpropto(I, \iota^{-1})$ similarly. Many nice properties hold for this correspondence; for instance, we have $\varpropto(\varpropto(I, \iota), \iota^{-1})$ is a subinterval of $I$ and there exist (possibly empty) finite subintervals $I_0, I_2 \subseteq I$ such that $I \equiv I_0 \varpropto(\varpropto(I, \iota), \iota^{-1}) I_2$.

If $W \in \textbf{Red}_H$ and $U \in \textbf{Red}_T$ and $\iota$ is a coi from $\text{p-index}(W)$ to $\text{p-index}(U)$ then we write $\text{coi}(W, \iota, U)$ and call such a triple a coi triple. A collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ of coi triples is coherent if for any choice of $x_0, x_1 \in X$, intervals $I_0 \subseteq \text{p-index}(W_{x_0})$ and $I_1 \subseteq \text{p-index}(W_{x_1})$, and $i \in \{-1, 1\}$ such that

$W_{x_0} \upharpoonright_p I_0 \equiv (W_{x_1} \upharpoonright_p I_1)^i$

we get that

$[[U_{x_0} \upharpoonright_p \varpropto(I_0, \iota_{x_0})]] = [[(U_{x_1} \upharpoonright_p \varpropto(I_1, \iota_{x_1}))^i]]$

and also for any choice of $x_2, x_3 \in X$, intervals $I_2 \subseteq \text{p-index}(U_{x_2})$ and $I_3 \subseteq \text{p-index}(U_{x_3})$, and $j \in \{-1, 1\}$ such that

$U_{x_2} \upharpoonright_p I_2 \equiv (U_{x_3}\upharpoonright_p I_3)^j$

we get that

$[[W_{x_2} \upharpoonright_p \varpropto(I_2, \iota_{x_2}^{-1})]] = [[(W_{x_3} \upharpoonright_p \varpropto(I_3, \iota_{x_3}^{-1}))^j]]$.

Note that it is possible that a collection of coi triples which has only one element can fail to be coherent, since the above definition allows that $x_0 = x_1$ and $I_0 \neq I_1$. Thus, great care must be taken in producing a coherent collection of coi triples. For a coherent example, one can take

$W \equiv a_0^1a_1^{-2}a_2^{3}a_3^{-4}a_4^{-5}\cdots$

and

$U \equiv t_{0, 0}t_{1, 1}t_{0, 2}t_{1, 3}t_{0, 4}\cdots$

where we clearly have $\text{p-index}(W) \equiv \mathbb{N} \equiv \text{p-index}(U)$ and let $\iota$ be the unique order isomorphism between $\text{p-index}(W)$ and $\text{p-index}(U)$. Of course, one needs to check that the appropriate conditions hold in order to conclude that $\{\text{coi}(W, \iota, U)\}$ is coherent. As a hint in this easy example, one can see that if $I_0, I_1 \subseteq \text{p-index}(W)$ are intervals and $W \upharpoonright_p I_0 \equiv W\upharpoonright_p I_1$ and either of $I_0$ or $I_1$ is finite, then both are finite and $[[U \upharpoonright_p \varpropto(I_0, \iota)]] = [[E]] = [[U \upharpoonright_p \varpropto(I_1, \iota)]]$. If at least one of $I_0$ or $I_1$ is infinite then both must be infinite and in fact $W \upharpoonright_p I_0 \equiv W\upharpoonright_p I_1$ implies $I_0 = I_1$ and the desired equality once again holds. One cannot have $W\upharpoonright_p I_0 \equiv (W\upharpoonright_p I_1)^{-1}$ if either of $I_0$ or $I_1$ is infinite (by considering the order type), and if either of $I_0$ or $I_1$ is finite we again see that $[[U \upharpoonright_p \varpropto(I_0, \iota)]] = [[E]] = [[(U \upharpoonright_p \varpropto(I_1, \iota))^{-1}]]$. The check for intervals in $\text{p-index}(U)$ is comparable.

One can imagine that the check for coherence becomes annoying when the collection has more elements and words become more complicated. The payoff for producing such a collection, however, is hinted at in the following:

Proposition. Let $\beth_T: \textbf{Red}_T \rightarrow \textbf{Red}_T/\langle\langle \textbf{Pure}_T\rangle\rangle$ and $\beth_H: \textbf{Red}_H \rightarrow \textbf{Red}_H/\langle\langle \textbf{Pure}_H \rangle\rangle$ denote the respective quotient maps. A coherent collection $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ of coi triples induces an isomorphism

$\Phi: \beth_H(\text{P-fine}(\{W_x\}_{x \in X})) \rightarrow \beth_T(\text{P-fine}(\{U_x\}_{x \in X}))$.

If we can produce a coherent collection of coi triples $\{\text{coi}(W_x, \iota_x, U_x)\}_{x \in X}$ which is plentiful enough that

$\beth_H(\text{P-fine}(\{W_x\}_{x \in X})) = \textbf{Red}_H/\langle\langle \textbf{Pure}_H\rangle\rangle$

and

$\beth_T(\text{P-fine}(\{U_x\}_{x \in X})) = \textbf{Red}_T/\langle\langle \textbf{Pure}_T \rangle\rangle$

then we have obtained an isomorphism.

Although the proposition is very intuitive, the proof is technical. One first shows that from a coherent collection one obtains well-defined homomorphisms $\text{P-fine}(\{W_x\}_{x\in X}) \rightarrow \textbf{Red}_T/\langle\langle \textbf{Pure}_T\rangle\rangle$ and $\text{P-fine}(\{U_x\}_{x\in X}) \rightarrow \textbf{Red}_H/\langle\langle \textbf{Pure}_H\rangle\rangle$. That these natural maps (extending $W\upharpoonright_p I \mapsto [[U\upharpoonright_p \varpropto (I, \iota)]]$) are well-defined requires some effort, since a word $W \in \text{P-fine}(\{W_x\}_{x\in X})$ might decompose in many distinct ways as a finitary concatenation of elements in $\bigcup_{x\in X} \text{p-chunk}(\{W_x^{\pm 1}\}_{x \in X})$, and coherence is essential to the argument. Once these homomorphisms are in hand, it is easier to see that they descend to homomorphisms $\beth_H(\text{P-fine}(\{W_x\}_{x \in X})) \rightarrow \textbf{Red}_T/\langle\langle \textbf{Pure}_T\rangle\rangle$ and $\beth_T(\text{P-fine}(\{U_x\}_{x \in X})) \rightarrow \textbf{Red}_H/\langle\langle \textbf{Pure}_H\rangle\rangle$ and that these homomorphisms are inverse to each other.

## The Griffiths twin cone and the harmonic archipelago have isomorphic fundamental group (Part 1)

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 $\textbf{TC}$ 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 $\textbf{HA}$ 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 $\pi_1(\textbf{TC}) \simeq \pi_1(\textbf{HA})$ originated with James W. Cannon and Gregory R. Conner.

Recall that the earring space $\textbf{E}$ is the shrinking wedge of countably infinitely many circles. More formally if $p \in \mathbb{R}^2$ we let $C(p, r)$ denote the circle centered at $p$ of radius $r$. The subspace $\textbf{E} \subseteq \mathbb{R}^2$ is given by $\textbf{E} = \bigcup_{n\in \mathbb{N}} C((0, \frac{1}{n + 1}), \frac{1}{n + 1})$ (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 $\pi_1(\textbf{E})$ will be like a free group. While this is true, it is emphasized that $\pi_1(\textbf{E})$ is not a free group. This is best illustrated by the curious fact that $| \pi_1(\textbf{E})| =2^{\aleph_0}$ and $\pi_1(\textbf{E})$ 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 $A = \{a_n^{\pm 1}\}_{n \in \mathbb{N}}$ be a countably infinite collection of symbols, which we will call letters, which is equipped with formal inverses. Usually the superscript $1$ is not written. A word $W$ is a finite-to-one function $W: \overline{W} \rightarrow A$ where the domain $\overline{W}$ is a totally ordered set (finite-to-one means in this case that for each $n \in \mathbb{N}$ and $\epsilon \in \{\pm 1\}$ the set $\{t \in \overline{W} : W(t) = a_n^{\epsilon}\}$ is finite). It follows that the domain $\overline{W}$ of a word $W$ must be countable (possibly finite or empty). As an example the infinite string

$a_0^{-1}a_1a_2^{-1}a_3a_4^{-1}a_5 \cdots$

is a word; more formally it is the word $W: \mathbb{N} \rightarrow A$ given by $W(n) = a_n^{{(-1)}^{n + 1}}$ (notice that each element of the alphabet $A$ is utilized at most once in the word). The infinite string

$a_0a_1a_0a_3a_0a_5 \cdots$

given by the rule $n\mapsto \begin{cases}a_0 \text{ if }n\text{ is even}\\a_n\text{ if }n\text{ is odd } \end{cases}$ is not a word since the letter $a_0$ is used infinitely often. Let $E$ denote the empty word, i.e. the word with empty domain. A word can have more exotic domain than $\mathbb{N}$: any finite-to-one function $W: \mathbb{Q} \rightarrow A$ is a word. As a technical aside, we consider two words $W_0$ and $W_1$ to be equivalent, and write $W_0 \equiv W_1$, provided there exists an order isomorphism $\iota: \overline{W_0} \rightarrow \overline{W_1}$ such that $W_0(t) = W_1(\iota(t))$ for all $t\in \overline{W_0}$. We form the concatenation of two words $W_0$ and $W_1$, denoted $W_0W_1$, by declaring that $W_0W_1$ has domain which is the disjoint union $\overline{W_0} \sqcup \overline{W_1}$ with the elements in $\overline{W_0}$ being ordered below those in $\overline{W_1}$ and having

$W_0W_1(t) = \begin{cases}W_0(t)\text{ if }t\in \overline{W_0}\\W_1(t)\text{ if }t\in \overline{W_1}\end{cases}$

Analogously, given a totally ordered set $\Lambda$ and collection of words $\{W_{\lambda}\}_{\lambda \in \Lambda}$ indexed by $\Lambda$ we can form a function whose domain is the disjoint union $\bigsqcup_{\lambda \in \Lambda}\overline{W_{\lambda}}$, ordered in the natural way, and defined by $t \mapsto W_{\lambda}(t)$ where $t \in \overline{W_{\lambda}}$. This function we denote $\prod_{\lambda} W_{\lambda}$ and it is a word provided it is finite-to-one.

A word $W$ has an inverse, which is denoted $W^{-1}$, given by letting $\overline{W^{-1}}$ be the set $\overline{W}$ under the reverse order and $W^{-1}(t) = (W(t))^{-1}$. For example the inverse of the word

$a_0^{-1}a_1a_2^{-1}a_3a_4^{-1}a_5 \cdots$

will be the word

$\cdots a_5^{-1}a_4a_3^{-1}a_2a_1^{-1}a_0$

Given $N \in\mathbb{N}$ and word $W$ we let $p_N(W)$ be the finite word given by the restriction $W \upharpoonright\{t\in \overline{W}: W(t) \in \{a_0^{\pm 1}, \ldots, a_N^{\pm 1}\}\}$. Given words $W_0, W_1$ we write $W_0 \sim W_1$ if for each $N \in \mathbb{N}$ the words $p_N(W_0)$ and $p_N(W_1)$ are equal as elements in the free group. For example, the word $W$

$a_0a_1^2a_4a_5a_6a_7a_8a_9 \cdots \cdots a_9^{-1}a_8^{-1}a_7^{-1}a_6^{-1}a_5^{-1}a_4^{-1}a_1^{-3}a_0$

has $p_0(W) \equiv a_0^2$, $p_1(W) \equiv a_0a_1^2a_1^{-3}a_0 \equiv p_2(W) \equiv p_3(W)$ and for $N \geq 4$ we get

$p_N(W) \equiv a_0a_1^2a_4 \cdots a_N a_N^{-1}\cdots a_4^{-1}a_1^{-3}a_0$

It is easy to see that $a_0a_1^{-1}a_0 \sim W$.

The group $\pi_1(\textbf{E})$ is isomorphic to the collection of equivalence classes over $\sim$. The binary operation is given by concatenation: $(W_0/\sim) * (W_1/\sim) = (W_0W_1)/\sim$ and the $\sim$ class of the empty word $E$ plays the role of the group identity. Inverses in the group are predictably defined by $(W/\sim)^{-1} = W^{-1}/\sim$.

Analogously to a free group, there are specific words with which we prefer to work. Given a word $W$ we say that $W_1$ is a subword of $W$ if there exist words $W_0, W_2$ (either or both of which may be empty) such that $W \equiv W_0W_1W_2$. Moreover $W_1$ is an initial (respectively terminal) subword provided $W_0$ (resp. $W_2$) in the above writing is empty. Finally a word $W$ is reduced if for every subword $W_1$ we have $W_1\sim E$ implies $W_1 \equiv E$. 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 $\sim$ class contains a reduced word which is unique up to $\equiv$. Letting $\textbf{Red}(W)$ denote the reduced representative of the $\sim$ class of word $W$ we have for all words $W_0, W_1, W_2$ that $\textbf{Red}(W_0 \textbf{Red}(W_1W_2)) \equiv \textbf{Red}(\textbf{Red}(W_0W_1)W_2)$. Moreover, given reduced words $W, W'$ there exist words $W_0, W_1, W_0', W_1'$ such that

(1) $W \equiv W_0W_1$;

(2) $W' \equiv W_0'W_1'$;

(3) $(W_1)^{-1} \equiv W_0'$;

(4) $W_0W_1'$ is reduced.

For further reading on (reduced) words see Section 1 of K. Eda, Free $\sigma$-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 $\textbf{Red}$ of reduced words with binary operation $W_0*W_1 \equiv \textbf{Red}(W_0W_1)$. We introduce two alphabets with formal inverses:

$H = \{h_n^{\pm 1}\}_{n \in \mathbb{N}}$ (with $H$ for “h”armonic archipelago); and

$T = \{t_{i, n}^{\pm 1}\}_{i \in \{0, 1\}, n \in \mathbb{N}}$ (with $T$ for “t”win cone).

Define words, concatenation, $\sim$, reduced word, etc. just as before for each of these new alphabets and let $\textbf{Red}_H$ and $\textbf{Red}_T$ denote the respective sets of reduced words. These two sets are each groups under the binary operation $W_0*W_1 \equiv \textbf{Red}(W_0W_1)$ and both are isomorphic to $\textbf{Red}$ (the isomorphism with $\textbf{Red}_H$ is given by the word mapping which extends $a_n^{\pm 1} \mapsto h_n^{\pm 1}$ and the isomorphism with $\textbf{Red}_T$ is given by $a_n^{\pm 1} \mapsto t_{i, m}^{\pm 1}$ where $n = 2m + i$).

A word $W \in \textbf{Red}_T$ is $(0, T)-$pure if the first subscript in each of the letters appearing in $W$ is $0$, and $(1, T)$pure is defined analogously. A word is $T$pure provided it is either $(0, T)$-pure or $(1, T)$-pure. For $i \in \{0, 1\}$ every subword of a $(i, T)$-pure word is again $(i, T)$-pure, and the only word which is both $(0, T)$-pure and $(1, T)$-pure is $E$. Let $\textbf{Pure}_T$ denote the set of $T$-pure words. The group $\pi_1(\textbf{TC})$ is isomorphic to $\textbf{Red}_T/\langle\langle \textbf{Pure}_T \rangle\rangle$, where the notation $\langle\langle \cdot \rangle\rangle$ 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 $W \in \textbf{Red}_H$ is $(n, H)$pure, where $n \in \mathbb{N}$, provided all subscripts of letters appearing in $W$ are $n$ (i.e. $W$ is of form $h_n^j$ where $j \in \mathbb{Z}$). A word is $H$pure provided it is $(n, H)$-pure for some $n \in \mathbb{N}$ and we let $\textbf{Pure}_H$ denote the set of $H$-pure words. The group $\pi_1(\textbf{HA})$ is isomorphic to $\textbf{Red}_H/\langle\langle \textbf{Pure}_H\rangle\rangle$ (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 $\pi_1(\textbf{TC}) \simeq \pi_1(\textbf{HA})$ is reduced to producing an isomorphism between $\textbf{Red}_T/\langle\langle \textbf{Pure}_T \rangle\rangle$ and $\textbf{Red}_H/\langle\langle \textbf{Pure}_H\rangle\rangle$. This is not an easy task. It’s a nice exercise to check that any continuous function $f: \textbf{TC} \rightarrow \textbf{HA}$ induces a trivial homomorphism $f_*: \pi_1(\textbf{TC}) \rightarrow \pi_1(\textbf{HA})$ (using the fact that $\textbf{TC}$ 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 $f: \textbf{HA} \rightarrow \textbf{TC}$ so that $f_*$ is surjective, it is not possible to make such an $f_*$ injective as well. Thus, the natural (spacial) homomorphisms are ruled out. The fact that each element of $\textbf{Red}_T$ is a (possibly infinitary) concatenation of $T$-pure words and similarly each element of $\textbf{Red}_H$ is a (possibly infinitary) concatenation of $H$-pure words should be used in some way. A confounding issue is that $|\textbf{Pure}_T| = 2^{\aleph_0}$ and $|\textbf{Pure}_H| = \aleph_0$. We will continue in Part 2.