Co-Hopfianity is not a profinite property

This paper demonstrates that co-Hopfianity is not a profinite property by constructing two finitely generated residually finite groups with isomorphic profinite completions, where one is co-Hopfian and the other is not, utilizing Wise's Rips construction and a specific preimage subgroup technique.

The Gist

Here is an explanation of the paper "Co-Hopfianity Is Not a Profinite Property" using simple language, analogies, and metaphors.

The Big Picture: The "Fingerprint" Problem

Imagine you have two mysterious boxes, Box G and Box H. You can't open them to see what's inside, but you can take a "fingerprint" of each box. In the world of mathematics, this fingerprint is called the profinite completion.

Think of the fingerprint as a collection of all the possible "shadows" the box casts when you shine a light through it. If you look at the box through a series of different, smaller, simpler lenses (finite quotients), you get a specific pattern. If two boxes cast the exact same pattern through every possible lens, mathematicians say their fingerprints are identical ($\hat{G} \cong \hat{H}$).

For a long time, mathematicians wondered: If two boxes have the exact same fingerprint, do they have to be the same kind of box inside? Specifically, do they share the same "personality traits" (algebraic properties)?

This paper says: No. Two boxes can have identical fingerprints but completely different internal personalities.

The Specific Trait: "Co-Hopfianity"

To prove this, the authors focus on a specific personality trait called Co-Hopfianity.

• The Analogy: Imagine a shape-shifting robot.
  • A Hopfian robot is one that cannot be squashed into a smaller version of itself without losing pieces. (If you push it, it stays the same size).
  • A Co-Hopfian robot is one that cannot hide inside itself. It cannot fit a perfect copy of itself inside its own body as a smaller, proper part.
  • A Non-Co-Hopfian robot is like a Russian nesting doll that is broken: you can find a perfect, smaller copy of the whole robot hiding inside its own chest.

The Goal: The authors wanted to build two robots, G and H, that look identical from the outside (same fingerprint), but:

1. Robot G is a "Co-Hopfian" robot (it cannot hide a copy of itself inside).
2. Robot H is a "Non-Co-Hopfian" robot (it can hide a copy of itself inside).

If they succeed, it proves that "Co-Hopfianity" is not a trait you can detect just by looking at the fingerprint.

How They Built the Robots

The authors used a clever construction method called the Rips Construction (invented by mathematician Daniel Wise). Think of this as a 3D printer that takes a simple blueprint and prints a complex, hyperbolic machine.

Step 1: The Master Blueprint (Group U)

They started with a very strange, special blueprint called Group U.

• It's "acyclic," meaning it has no holes or loops in its structure (mathematically, its homology is zero).
• It has a trivial fingerprint. This means if you look at it through any lens, it looks like empty space. It's a "ghost" blueprint.
• Crucially, it has a "universality" property: you can build almost any other machine inside it.

Step 2: Building Robot G (The Rigid One)

They used the Rips printer to build Robot G based on blueprint U.

• Result: Robot G is a "torsion-free hyperbolic group."
• The Analogy: Imagine a rigid, one-piece diamond structure. It's so tightly locked together that you cannot fit a smaller, perfect copy of the whole diamond inside itself.
• Why? A famous theorem by Z. Sela says that if a structure is "one-ended" (it doesn't split into two separate pieces) and has no loose parts, it is rigid. Robot G fits this description perfectly. G is Co-Hopfian.

Step 3: Building Robot H (The Flexible One)

Now, they needed to build Robot H using the same blueprint U, but with a twist.

• Inside the ghost blueprint U, they found a special "pocket" (Subgroup A) that looks exactly like U itself.
• They also found a "magic key" (Element $t$) that, when used, shrinks that pocket. It turns the pocket into a smaller version of itself ($t^{-1}At \subsetneq A$).
• They built Robot H by taking the original machine G and only keeping the parts that correspond to this shrinking pocket.
• The Trick: Because the pocket shrinks inside the blueprint, the machine H can be "conjugated" (rotated/moved by the magic key) to fit perfectly inside a smaller part of itself.
• Result: Robot H can hide a copy of itself inside its own body. H is NOT Co-Hopfian.

The Grand Reveal: Same Fingerprint, Different Robots

Here is the magic trick that ties it all together:

1. The Ghost Effect: Because the blueprint U has a "trivial fingerprint" (it looks like nothing), the "shadows" cast by the original machine G and the "pocket" machine H are identical.
2. The Mathematical Proof: Using a theorem by Bridson and Grunewald, the authors showed that because the blueprint U is a "ghost," the fingerprint of G is exactly the same as the fingerprint of the "kernel" (the core part), which is also the same as the fingerprint of H.
3. The Conclusion:
   • $\hat{G} \cong \hat{H}$ (They look identical from the outside).
   • $G$ is rigid (Co-Hopfian).
   • $H$ is flexible (Not Co-Hopfian).

Why This Matters

Before this paper, mathematicians knew that some traits (like being "abelian" or "nilpotent") could be detected by the fingerprint. They also knew that some geometric traits (like "amenability") could not.

This paper adds Co-Hopfianity to the list of traits that cannot be detected by the fingerprint. It shows that even if you have a perfect, complete list of all the finite "shadows" a group can cast, you still cannot tell if that group is rigid enough to prevent itself from hiding inside itself.

In short: You can have two groups that are indistinguishable by all their finite parts, yet one is a solid, unbreakable diamond, and the other is a set of nesting dolls that can fold into itself. The fingerprint doesn't tell the whole story.

Technical Summary

Here is a detailed technical summary of the paper "Co-Hopfianity is not a profinite property" by Hyungryul Baik and Wonyong Jang.

1. Problem Statement

The central problem addressed in this paper is the profinite rigidity of group properties. Specifically, the authors investigate whether the co-Hopfian property is a profinite property.

• Definition of Co-Hopfian: A group $G$ is co-Hopfian if every injective homomorphism $\phi: G \to G$ is surjective. Equivalently, $G$ is not isomorphic to any of its proper subgroups.
• Definition of Profinite Property: A property $P$ is a profinite property if, for any two finitely generated residually finite groups $G_1$ and $G_2$ with isomorphic profinite completions ($\widehat{G_1} \cong \widehat{G_2}$), $G_1$ has property $P$ if and only if $G_2$ has property $P$.
• Context: While many algebraic properties (e.g., being abelian, nilpotent) are known to be profinite, many geometric properties (e.g., amenability, property (T)) are not. The co-Hopfian property is known to be delicate; for instance, while finitely generated residually finite groups are always Hopfian, they are not necessarily co-Hopfian. The authors aim to prove that co-Hopfianity is not determined by the profinite completion.

2. Methodology

The authors construct two specific finitely generated residually finite groups, $G$ and $H$, satisfying three conditions:

1. $\widehat{G} \cong \widehat{H}$ (Isomorphic profinite completions).
2. $G$ is co-Hopfian.
3. $H$ is not co-Hopfian.

The construction relies on three main mathematical tools:

A. Bridson's Universal Acyclic Group ($U$)

The authors utilize a finitely presented group $U$ constructed by Bridson [5] with the following critical properties:

• Acyclic: $H_i(U, \mathbb{Z}) = 0$ for all $i \geq 1$.
• Trivial Profinite Completion: $\widehat{U} = 1$.
• Universality: Every finitely presented group embeds into $U$.

B. Wise's Residually Finite Rips Construction

They apply a variation of the Rips construction (due to Wise [25]) to the group $U$. This yields a short exact sequence:
$$1 \to K \to G \xrightarrow{\pi} U \to 1$$
where:

• $G$ is a torsion-free, hyperbolic, residually finite group.
• $K$ is a finitely generated kernel (generated by 3 elements).
• Because $U$ is acyclic and has trivial profinite completion, Lemma 2.5 (Bridson–Grunewald) implies that the inclusion induces an isomorphism $\widehat{K} \cong \widehat{G}$.

C. The "Self-Embedding" via Conjugation

To construct the non-co-Hopfian group $H$, the authors exploit the universality of $U$:

1. They construct a subgroup $A < U$ and an element $t \in U$ such that $A \cong U$ but $t^{-1}At \subsetneq A$ (a proper inclusion).
   • This is achieved by embedding $U * F_2$ into $U$ via a map $\psi$, defining a non-surjective endomorphism $\sigma$, and using an HNN extension to embed the result back into $U$.
2. They define $H$ as the preimage of $A$ under the projection $\pi$:
   $$H := \pi^{-1}(A) < G$$
3. They show that conjugation by a lift $g \in G$ (where $\pi(g)=t$) maps $H$ isomorphically onto a proper subgroup of itself ($\pi^{-1}(t^{-1}At)$).

3. Key Results and Proofs

Theorem 3.1 (Main Result)

There exist finitely generated residually finite groups $G$ and $H$ such that $\widehat{G} \cong \widehat{H}$, $G$ is co-Hopfian, and $H$ is not.

Proof of Co-Hopfianity for $G$:

• $G$ is a torsion-free hyperbolic group.
• To apply Sela's theorem (which states torsion-free one-ended hyperbolic groups are co-Hopfian), the authors prove $G$ is one-ended.
• If $G$ were not one-ended, it would split as a non-trivial free product $G_1 * G_2$. By a classical result, a non-trivial finitely generated normal subgroup (like $K$) in such a split must have finite index. However, $G/K \cong U$ is infinite, leading to a contradiction. Thus, $G$ is one-ended and co-Hopfian.

Proof of Non-Co-Hopfianity for $H$:

• $H = \pi^{-1}(A)$.
• Since $t^{-1}At \subsetneq A$, the preimage satisfies $\pi^{-1}(t^{-1}At) \subsetneq \pi^{-1}(A) = H$.
• Conjugation by $g$ (where $\pi(g)=t$) restricts to an isomorphism $c_g|_H: H \to \pi^{-1}(t^{-1}At)$.
• Therefore, $H$ is isomorphic to a proper subgroup of itself.

Proof of Isomorphic Profinite Completions:

• For $G$: The sequence $1 \to K \to G \to U \to 1$has$\widehat{U}=1$and$H_2(U, \mathbb{Z})=0$(since$U$is acyclic). By Lemma 2.5,$\widehat{K} \cong \widehat{G}$.
• For $H$: The sequence $1 \to K \to H \to A \to 1$exists. Since$A \cong U$,$\widehat{A}=1$and$H_2(A, \mathbb{Z})=0$. Thus,$\widehat{K} \cong \widehat{H}$.
• Conclusion: $\widehat{G} \cong \widehat{K} \cong \widehat{H}$.

4. Additional Contribution: Toral Relative Hyperbolicity

As a side result (Proposition 2.9), the authors demonstrate that toral relative hyperbolicity is also not a profinite property.

• $G$ is torsion-free hyperbolic, hence toral relatively hyperbolic.
• $K$ (the kernel) is not finitely presented (a standard property of the Rips construction when the quotient is infinite), whereas any toral relatively hyperbolic group that is a subgroup of a hyperbolic group must be finitely presented (to avoid containing $\mathbb{Z}^2$).
• Thus, $G$ has the property, but $K$ (which has $\widehat{K} \cong \widehat{G}$) does not.

5. Significance

• Resolution of an Open Question: The paper definitively answers whether co-Hopfianity is a profinite property in the negative. This adds to the list of geometric and asymptotic properties that cannot be detected solely by finite quotients.
• Refinement of Rigidities: It highlights the limitations of profinite rigidity. While some classes of groups (like virtually compact special groups) exhibit rigidity regarding relative hyperbolicity, this does not hold for general finitely generated residually finite groups.
• Methodological Innovation: The construction of $H$ as a preimage subgroup that is conjugate to a proper subgroup of itself provides a clean, structural mechanism for generating non-co-Hopfian groups without requiring deep analysis of the Rips kernel's internal structure. This approach leverages the "universality" of Bridson's group to create the necessary self-embedding.

In summary, Baik and Jang successfully decouple the co-Hopfian property from the profinite completion, showing that two groups can share all their finite quotients yet differ fundamentally in their internal subgroup structure regarding self-embeddings.