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.

This entry was posted in Fundamental group, Griffiths twin cone, harmonic archipelago, Infinite Group Theory, Infinite products, Order Theory and tagged , , . Bookmark the permalink.

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