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


\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.

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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


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


\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.

Posted in Fundamental group, Griffiths twin cone, harmonic archipelago, Infinite Group Theory, Infinite products, Order Theory | Tagged , , | Leave a comment

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 Tpure 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 Hpure 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.

Posted in Fundamental group, Griffiths twin cone, Group theory, harmonic archipelago, Infinite Group Theory | Tagged | 2 Comments