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.