Diamonds: Homology and the Central Series of Groups

1. Problem Statement

The paper addresses a fundamental question in group theory and geometric topology: To what extent is a group $\Gamma$ determined by its homomorphisms into a specific class of groups $\mathcal{C}$?

Specifically, the authors focus on the class of nilpotent groups. A group $\Gamma$ is residually nilpotent if the intersection of its lower central series terms is trivial. The "nilpotent genus" of such a group is the number of isomorphism classes of finitely presented, residually nilpotent groups that share the same sequence of universal nilpotent quotients $N_j(\Gamma) = \Gamma/\Gamma_j$ for all $j$.

The central problem is whether the sequence of nilpotent quotients uniquely determines the group (or the associated geometric object, such as a manifold or number field). While John Stallings (1965) proved that a homomorphism $\psi: \Gamma \to \Lambda$ inducing isomorphisms on $H_1$ and surjections on $H_2$ induces isomorphisms on all nilpotent quotients, this paper investigates a scenario where no direct homomorphism exists between the two groups in question. Instead, the groups are subgroups of a common ambient group, and the isomorphism of quotients is derived via a "diamond" configuration of homological data.

2. Methodology

The authors develop a novel homological framework to construct pairs of non-isomorphic groups with isomorphic nilpotent quotients.

A. The "Diamond" Theorem (Main Technical Tool)

The core theoretical contribution is an analog to Stallings' Theorem adapted for a "diamond" of subgroups.

• Setup: Let $\Omega$ be a group containing subgroups $\Gamma$ and $\Lambda$.
• Condition: The authors require that the inclusion maps induce isomorphisms on the first two homology groups ($H_1$ and $H_2$) with integer coefficients, and that these isomorphisms commute with the maps from the ambient group $\Omega$ and the intersection $\Gamma \cap \Lambda$.
  $$\psi_k: H_k(\Gamma, \mathbb{Z}) \xrightarrow{\cong} H_k(\Lambda, \mathbb{Z})$$
  such that the diagrams involving $H_k(\Omega, \mathbb{Z})$ and $H_k(\Gamma \cap \Lambda, \mathbb{Z})$ commute for $k=1, 2$.
• Result (Theorem 1.9): Under these conditions, there exists an isomorphism $\eta_j: N_j(\Gamma) \to N_j(\Lambda)$ for all $j \geq 0$, making the diagram of nilpotent quotients commute.

B. Proof Strategy

The proof proceeds by induction on the nilpotency step $j$:

1. Base Case ($j=1$): $N_1(\Gamma) \cong H_1(\Gamma, \mathbb{Z})$. The hypothesis provides the isomorphism directly.
2. Inductive Step: The authors utilize the Lyndon–Hochschild–Serre (LHS) spectral sequence to relate the homology of the group to its central extensions.
   • They analyze the 5-term exact sequence associated with the extension $1 \to \Gamma_j/\Gamma_{j+1} \to N_{j+1}(\Gamma) \to N_j(\Gamma) \to 1$.
   • Using Lemma 2.4, they establish that if the "diamond" of central extensions (involving $\Gamma, \Lambda, \Omega, \Gamma \cap \Lambda$) has compatible isomorphisms on the base and kernel terms, a unique isomorphism exists for the total extension groups ($N_{j+1}$).
   • This allows them to lift the isomorphism from $N_j$ to $N_{j+1}$ without requiring a direct map $\Gamma \to \Lambda$.

C. Construction of Examples

To apply this theorem, the authors utilize the concept of $\mathbb{Z}$-coset equivalence. Two subgroups $\Gamma, \Lambda \leq \Omega$ are $\mathbb{Z}$-coset equivalent if the permutation modules $\mathbb{Z}[\Omega/\Gamma]$ and $\mathbb{Z}[\Omega/\Lambda]$ are isomorphic as $\mathbb{Z}[\Omega]$-modules.

• By Shapiro's Lemma, $\mathbb{Z}$-coset equivalence implies isomorphisms on all homology groups $H_k(\Gamma, \mathbb{Z}) \cong H_k(\Lambda, \mathbb{Z})$ that commute with the ambient maps.
• The authors construct specific examples of non-conjugate, $\mathbb{Z}$-coset equivalent subgroups within non-arithmetic lattices of Lie groups.

3. Key Contributions and Results

A. Hyperbolic Geometry (Real and Complex)

• Theorem 1.1 & 1.3: The authors construct pairs of non-isometric, finite covers of compact (or finite volume) non-arithmetic hyperbolic $n$-orbifolds ($n \geq 3$ for real, $n=2$ for complex) such that their fundamental groups are residually nilpotent and have isomorphic universal nilpotent quotients.
• Corollary 1.2 & 1.4: For any integer $\ell > 1$, there exist closed hyperbolic manifolds (real and complex) with nilpotent genus greater than $\ell$.
• Significance: These are the first examples of residually nilpotent Kleinian groups with arbitrarily large nilpotent genus. Prior to this, no examples of non-isomorphic residually nilpotent Kleinian groups with isomorphic nilpotent completions were known.
• Invariants: These manifolds share many invariants: volume, Euler characteristic, Laplacian spectrum, geodesic length spectrum, and integral K-theory.

B. Number Theory (Galois Groups)

• Corollary 1.5: The authors construct non-isomorphic number fields $K_1, K_2$ whose absolute Galois groups $\Gamma_{K_1}$ and $\Gamma_{K_2}$ have isomorphic universal nilpotent quotients.
• Significance: This demonstrates that nilpotent representations are not anabelian. While Uchida (1970s) showed that number fields are determined by their solvable quotients (specifically 3-step), this result proves that the information is lost when restricting to nilpotent quotients.

C. Algebraic Geometry (Anabelian Geometry)

• Corollary 1.6 & 1.7: Considering curves and surfaces defined over a number field $k$, the authors show that the outer action of the absolute Galois group $Gal(\bar{k}/k)$ on the universal nilpotent quotients of the geometric fundamental group does not determine the curve or surface.
• Contrast: This stands in sharp contrast to Mochizuki's work, which proved that the outer action on the full geometric fundamental group (profinite completion) determines a hyperbolic curve.
• Conclusion: The "nilpotent representation theory" of geometric fundamental groups is insufficient for anabelian reconstruction.

4. Significance and Implications

1. Refutation of Nilpotent Anabelian Conjectures: The paper definitively shows that the "nilpotent anabelian" program fails. One cannot recover a number field, a hyperbolic curve, or a hyperbolic manifold solely from the system of their nilpotent quotients and the associated Galois actions.
2. New Class of Counterexamples: The construction provides a robust method (via $\mathbb{Z}$-coset equivalence in non-arithmetic lattices) to generate infinite families of non-isomorphic groups that are indistinguishable by nilpotent invariants.
3. Profinite Rigidity Context: The results clarify the boundary between "profinite rigidity" (determined by all finite quotients) and "nilpotent rigidity." While some groups are profinitely rigid, they may share the same nilpotent quotients with non-isomorphic groups.
4. Technical Innovation: The "Diamond Theorem" offers a powerful new tool for comparing groups that do not map to each other directly, relying instead on their shared embedding in a larger structure and homological compatibility.

5. Limitations and Open Questions

• The construction relies heavily on the existence of non-arithmetic and large lattices. The authors note that while non-arithmeticity is likely not strictly necessary, it simplifies the proof of non-conjugacy.
• The relationship between the profinite completions ($\widehat{\Gamma}$ and $\widehat{\Lambda}$) of the constructed groups remains an open question. The authors note that while the groups share nilpotent quotients, their profinite completions might differ (and recent work by Karshon et al. suggests they often do).
• The "nilpotent genus" for free groups and surface groups is known to be 1 (unique), but the behavior for general hyperbolic 3-manifolds remains an active area of investigation.

In summary, this paper establishes that the lower central series of a group (and its associated Galois actions) is a significantly coarser invariant than the full profinite completion, failing to distinguish between distinct geometric and arithmetic objects in high dimensions.