Griffiths Double Cone Space

• Griffiths Double Cone Space is defined as the union of two cones on the Hawaiian earring, joined along a common arc, producing a wild fundamental group.
• The space employs infinite reduced words and normal closures of pure subwords to model its fundamental group, resolving the Cannon–Conner conjecture.
• It exemplifies a broader class of wedge-like cone spaces where non-constructive isomorphisms reveal deep links between combinatorial group theory and topology.

Searching arXiv for papers on the Griffiths double cone space, harmonic archipelago, and related fundamental-group results. The Griffiths double cone space, often denoted $G$, is the union of two cones on the Hawaiian earring glued along the arc corresponding to the common basepoint segment. In the formulation given in "The double cone group is isomorphic to the archipelago group" (Corson, 15 Sep 2025), it is the “double cone over the infinite earring,” with fundamental group

$$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$

where the quotient kills all reduced words supported entirely in the $a$-letters or entirely in the $b$-letters. A central result is the resolution of the Cannon–Conner conjecture: the fundamental group of the Griffiths double cone is isomorphic to that of the harmonic archipelago (Corson, 15 Sep 2025). Related work further places the double cone in a broader family of $\kappa$-fold cone spaces whose fundamental groups are all isomorphic for $2\le \kappa\le 2^{\aleph_0}$, although these isomorphisms are purely non-constructive and are not induced by continuous maps (Corson, 2020).

1. Geometric construction and local structure

The underlying input space is the Hawaiian earring $\mathcal{E}$, described as the union of circles $C_n$ of radius $1/(n+1)$ tangent to the origin $o=(0,0)$ in $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$0 (Corson, 15 Sep 2025). For a based space $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$1 with basepoint $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$2, its reduced cone is

$$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$3

obtained by contracting $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$4 to a cone-point and collapsing $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$5 to a single segment (Corson, 15 Sep 2025). Since a cone over a connected space is contractible, $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$6 is contractible.

Griffiths’ space $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$7 is formed by taking two copies of $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$8 and identifying their copies of the “neck” $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$9 to a single arc. Equivalently,

$a$0

(Corson, 15 Sep 2025). This space is path-connected and locally path-connected except at the two cone-points; its wild behavior is concentrated at $a$1 (Corson, 15 Sep 2025).

A related presentation appears in the $a$2-fold construction of Corson, where $a$3 is the topological cone on the Hawaiian earring embedded as a Peano continuum in $a$4 with apex $a$5, and $a$6 is obtained by wedging $a$7 disjoint copies of $a$8 at the common basepoint $a$9 (Corson, 2020). The Griffiths double cone is the case $b$0.

2. Combinatorial model of the fundamental group

A major feature of the Griffiths double cone is that its fundamental group is most naturally expressed in terms of infinite reduced words. In the double-cone presentation one uses alphabets

$b$1

(Corson, 15 Sep 2025). A possibly infinite word is a finite-to-one map from a countable totally ordered set into $b$2, modulo order-preserving reparametrization of the domain. Reduction is defined by cancelling adjacent inverse letters in a maximal cancellation scheme, and $b$3 denotes the set of all reduced words (Corson, 15 Sep 2025).

Within this framework, a reduced word is called $b$4-pure if all letters lie in $b$5, $b$6-pure if all letters lie in $b$7, and pure if it is either $b$8-pure or $b$9-pure. The set of pure words is denoted $\kappa$0 (Corson, 15 Sep 2025). The fundamental group is then identified as

$\kappa$1

(Corson, 15 Sep 2025). Equivalently, one may regard $\kappa$2 as generated by the infinite families $\kappa$3 and $\kappa$4, subject only to the relations that every word in the subgroup generated by the $\kappa$5 alone is trivial and every word in the subgroup generated by the $\kappa$6 alone is trivial (Corson, 15 Sep 2025).

The $\kappa$7-fold analogue uses the alphabet

$\kappa$8

with words required to satisfy “finiteness at each level”: for each $\kappa$9, only finitely many letters have second index $2\le \kappa\le 2^{\aleph_0}$0 (Corson, 2020). Two such words are identified if all finite-level truncations represent the same element in the free group on $2\le \kappa\le 2^{\aleph_0}$1, producing a group $2\le \kappa\le 2^{\aleph_0}$2, and one then quotients by the normal closure of pure words supported in a fixed first index $2\le \kappa\le 2^{\aleph_0}$3 to obtain the cone group $2\le \kappa\le 2^{\aleph_0}$4 (Corson, 2020). For $2\le \kappa\le 2^{\aleph_0}$5, this gives the combinatorial model of the Griffiths double cone group.

This presentation is significant because it captures the non-semilocally-simply-connected behavior of the space through infinite-word combinatorics rather than through a finite CW-type model. A plausible implication is that the algebraic complexity of $2\le \kappa\le 2^{\aleph_0}$6 is inseparable from the infinite accumulation inherent in the Hawaiian earring.

3. Relation to the harmonic archipelago

The harmonic archipelago $2\le \kappa\le 2^{\aleph_0}$7 is another wild space whose fundamental group admits a parallel infinite-word description. In the account of Corson, $2\le \kappa\le 2^{\aleph_0}$8 is obtained from the unit disk by attaching infinitely many “tall hills” of height $2\le \kappa\le 2^{\aleph_0}$9 whose bases shrink to a boundary point $\mathcal{E}$0 (Corson, 15 Sep 2025). Its fundamental group is modeled using the alphabet

$\mathcal{E}$1

reduced words $\mathcal{E}$2, and pure words $\mathcal{E}$3, described there as all words supported in $\mathcal{E}$4, i.e. single-letter words $\mathcal{E}$5 (Corson, 15 Sep 2025). One obtains

$\mathcal{E}$6

(Corson, 15 Sep 2025).

The same group is also expressed using the topologist’s product: $\mathcal{E}$7 defined as the subgroup of the inverse limit $\mathcal{E}$8 consisting of infinite words with only finitely many occurrences of each generator, and the archipelago group

$\mathcal{E}$9

(Corson, 15 Sep 2025). One proves that $C_n$0.

The Cannon–Conner conjecture, stated in 1998, asserted that $C_n$1 is isomorphic to $C_n$2 (Corson, 15 Sep 2025). Theorem A of Corson’s paper establishes exactly this: $C_n$3 (Corson, 15 Sep 2025). In combinatorial terms,

$C_n$4

(Corson, 15 Sep 2025).

This identification is striking because the two spaces have substantially different geometric constructions. The result shows that, at the level of fundamental groups, the double cone and the harmonic archipelago lie in the same isomorphism class despite their differing ambient descriptions.

4. Proof strategy and non-constructive isomorphism

The proof of the isomorphism between the Griffiths double cone group and the harmonic archipelago group is combinatorial and non-constructive. The central device is a back-and-forth construction based on coherent families of close-order-isomorphism triples, abbreviated “coi” in the source (Corson, 15 Sep 2025).

The construction begins from the $C_n$5-decomposition of reduced words. In the $C_n$6-cone setting, every reduced word $C_n$7 decomposes uniquely as

$C_n$8

where each maximal subword $C_n$9 is pure, meaning that all its letters have the same first index $1/(n+1)$0 (Corson, 2020). The index set $1/(n+1)$1 is the $1/(n+1)$2-index of $1/(n+1)$3, and its intervals determine the $1/(n+1)$4-chunks (Corson, 2020). In the archipelago setting, one again has a $1/(n+1)$5-decomposition in $1/(n+1)$6 into maximal subwords lying in $1/(n+1)$7 (Corson, 2020).

A coherent coi-triple has the form

$1/(n+1)$8

where $1/(n+1)$9 and $o=(0,0)$0 are reduced words in the two alphabets and $o=(0,0)$1 is a close order-isomorphism between their $o=(0,0)$2-decomposition index sets (Corson, 15 Sep 2025). Coherence means, in essence, that matching $o=(0,0)$3-chunks agree compatibly across the entire family: whenever one $o=(0,0)$4-chunk of some $o=(0,0)$5 coincides, up to inverse, with a $o=(0,0)$6-chunk of another $o=(0,0)$7, the corresponding $o=(0,0)$8-chunks of $o=(0,0)$9 and $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$00 also coincide, and conversely (Corson, 2020).

The proof then proceeds by transfinite recursion of length continuum. The account in (Corson, 15 Sep 2025) states that any finite or countable union of coherent coi-triples is coherent, that coherent families can be extended to cover any new $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$01 or $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$02 by simple “small” or “$$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$03-type” or “$$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$04-type” concatenations, and that this eventually yields mutually inverse homomorphisms

$$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$05

$$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$06

(Corson, 15 Sep 2025). The 2020 preprint describes the same method as a back-and-forth over $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$07 many steps, alternating between well-orders of the relevant word sets and extending the coherent system by “arbitrary extensions” (Corson, 2020).

The sources explicitly note that the isomorphisms are non-constructive and use Zorn’s Lemma and transfinite induction (Corson, 15 Sep 2025). They also state that no continuous map $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$08 or $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$09 can induce the isomorphism (Corson, 15 Sep 2025). This rules out a common misconception: the equality of fundamental groups does not arise from an evident geometric equivalence of the spaces.

5. $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$10-fold generalizations and the size of the isomorphism class

The Griffiths double cone sits inside a larger family of wedge-like cone spaces. For any cardinal $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$11, the space $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$12 is formed from $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$13 copies of the cone on the Hawaiian earring by identifying all basepoints (Corson, 2020). When $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$14 is finite this is the wedge of $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$15 cones, while for $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$16 it is not compact but remains metrizable (Corson, 2020).

A principal theorem states that for all $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$17 with $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$18,

$$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$19

(Corson, 2020). The cardinality computation underlying the back-and-forth argument is

$$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$20

(Corson, 2020). In particular, for $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$21 one has $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$22 (Corson, 2020).

Corson’s later paper enlarges the isomorphism class even further. Theorem B there states that all of the following are isomorphic to $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$23: $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$24 for any $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$25, $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$26, and $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$27 for any sequence $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$28 of groups with $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$29 and no involutions (Corson, 15 Sep 2025). The source concludes that the isomorphism class is “very large,” containing uncountably many exotic descriptions (Corson, 15 Sep 2025).

This suggests that the Griffiths double cone group is better understood as a canonical representative of an extensive class of archipelago-type quotient groups rather than as an isolated fundamental group attached to one specific space.

6. Group-theoretic properties and structural consequences

The large isomorphism class of the Griffiths double cone group comes with a list of group-theoretic properties. According to (Corson, 15 Sep 2025), every such group is locally free, meaning that every finitely generated subgroup is free. It is also uncountable, contains every countable locally free group as a subgroup, and has abelianization

$$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$30

(Corson, 15 Sep 2025). The automorphism group has size at least $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$31 (Corson, 15 Sep 2025).

These properties are consistent with the infinite-word presentation. Local freeness reflects the fact that finite fragments of the structure behave like free-group data, while the quotient by normal closures of pure subwords only manifests its full effect at the infinite level. The abelianization formula exhibits a quotient of the full product by the finite-support direct sum, a familiar indicator of asymptotic rather than finite combinatorics.

Another structural formulation is the archipelago-group definition

$$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$32

where $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$33 denotes the topologist’s product (Corson, 15 Sep 2025). For $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$34, this recovers $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$35 and hence $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$36 (Corson, 15 Sep 2025). In this sense, the Griffiths double cone group occupies a nexus between wild fundamental groups and generalized topologist’s-product constructions.

7. Continuous-mapping obstructions and related misconceptions

A natural misunderstanding is to suppose that isomorphic fundamental groups should be induced by geometric maps between the spaces involved. The cited work explicitly excludes this. For the comparison among $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$37-fold cones, if $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$38 with $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$39, then

$$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$40

is uncountable; dually, any map $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$41 has uncountable kernel on fundamental groups (Corson, 2020). The argument uses the fact that loops can be homotoped into arbitrarily small neighborhoods of the wedgepoint, causing their images to lie in contractible cone-tip regions or forcing the collapse of many $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$42-chunk directions (Corson, 2020).

The same principle appears in the double-cone/archipelago comparison: no continuous map $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$43 or $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$44 can induce the isomorphism of fundamental groups (Corson, 15 Sep 2025). The proof of the isomorphism is therefore “purely combinatorial and non-constructive” (Corson, 15 Sep 2025).

The sources also note a broader contextual limitation. Kent’s planar-continuum theorem shows that in the plane $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$45-isomorphism often forces homotopy equivalence, but the Griffiths double cone and harmonic archipelago are not reducible to planar homotopy-equivalences (Corson, 15 Sep 2025). This situates the double cone among spaces for which algebraic invariants remain highly informative but do not rigidly determine geometric realization.

The Griffiths double cone space thus serves as a paradigmatic example of a wild continuum whose fundamental group admits multiple, radically different descriptions: as a normal-closure quotient of infinite reduced words in two alphabets, as the same group arising from the harmonic archipelago, and as one instance in a broad $$\pi_1(G)\cong \mathrm{Red}_{a,b}/\langle\!\langle \mathrm{Pure}_{a,b}\rangle\!\rangle,$$46-fold and archipelago-group isomorphism class. Its significance lies not only in the resolution of the Cannon–Conner conjecture, but also in the demonstration that for such spaces the relationship between topology, infinite combinatorics, and group structure is far less geometric than classical intuition might suggest (Corson, 15 Sep 2025).