Weak Commutativity Condition (WCC)

• The Weak Commutativity Condition (WCC) is a group construction imposing only diagonal commutator relations between a group and its isomorphic copy, preserving key structural properties.
• It leverages subgroups like L(H), D(H), and R(H) to analyze and retain polycyclic, nilpotent, and finite characteristics in algebraic frameworks.
• WCC integrates with non-abelian tensor squares and crossed modules, extending its impact on homological invariants and computational group theory.

The Weak Commutativity Condition (WCC) delineates a spectrum of algebraic and categorical constructions where certain minimal or diagonal commutator relations are enforced, as opposed to global commutativity. The paradigm is typified by the construction of the weak-commutativity group $\chi(H)$ associated to a group $H$ and an isomorphic copy $H^\psi$, requiring only that each $h \in H$ commutes with $h^\psi$ in the ambient group $\chi(H)$, not across all pairs of elements. This condition has deep ramifications for group theory, homological algebra, Lie theory, and computational group theory.

1. Fundamental Definition and Construction

Let $H$ be a group and let $\psi: H \to H^\psi$ be an isomorphism onto a second copy. The weak commutativity group is defined as

$$\chi(H) = \langle H \cup H^\psi \mid [h, h^\psi] = 1\ \forall h \in H \rangle,$$

with $[x, y] = x^{-1} y^{-1} x y$. The only mixed relations enforced are that each $h \in H$ commutes with its image $h^\psi$. No further commutations are imposed between arbitrary elements of $H$ and $H^\psi$. The construction is motivated by the desire to analyze the effect of minimal commutator constraints and to investigate the class of groups and functorial constructions preserved under such "diagonal" commutativity.

The operator $\chi$ preserves certain established properties:

• Finite $H$ implies finite $\chi(H)$
• Soluble $H$ of derived length $d$ gives $\chi(H)$ of derived length at most $d+1$
• Finitely generated nilpotent $H$ of class $c$ yields $\chi(H)$ nilpotent of class at most $2c$

Key normal subgroups are utilized:

• $L(H) = \ker(\chi(H) \to H)$, with $h \mapsto h$ and $h^\psi \mapsto h$
• $D(H) = \ker(\chi(H) \to H \times H)$, with $h \mapsto (h,1)$, $h^\psi \mapsto (1, h)$

These subgroups commute, and further extension analytic diagrams involving $W(H) = D(H) \cap L(H)$ and $R(H) = [H, L(H), H]$ play a role in structural analysis.

2. Preservation of Polycyclic and Polycyclic-by-Finite Structure

The principal theorem establishes:

• If $H$ is polycyclic, then $\chi(H)$ is polycyclic
• If $H$ is polycyclic-by-finite, then $\chi(H)$ is polycyclic-by-finite

The proof consists of several structural steps:

• Identification and analysis of subgroups $D(H)$, $L(H)$, $W(H)$, and $R(H)$, showing $[D(H), L(H)] = 1$, $R(H) \triangleleft \chi(H)$, $R(H) \subseteq W(H) \subseteq D(H)$, and $W(H)/R(H)$ is abelian.
• Construction of an explicit epimorphism $$\varepsilon: \chi(H) \twoheadrightarrow I_2(H) \times \{1\}$$, where $$I_2(H) = \langle (h-1)^2 \rangle \subset \ker(\varepsilon: \mathbb{Z}[H] \to \mathbb{Z})$$ and $I_2(H)$ is finitely generated for polycyclic-by-finite $H$.
• Application of the Schur–Huppert theorem ensuring that extensions of polycyclic-by-finite groups by finitely generated abelian central subgroups remain polycyclic-by-finite; thus, $L(H)$ is in the class, and so is $\chi(H)$.

Property of $H$	Corresponding Property of $\chi(H)$
polycyclic	polycyclic
polycyclic-by-finite	polycyclic-by-finite

This result robustifies the preservation paradigm for $\chi$ with respect to polycyclic classes, adding to the already-known retention of finiteness, solubility, and nilpotency.

3. Connections to Non-Abelian Tensor Squares and Crossed Modules

The construction of $\chi(H)$ facilitates new results for related functorial group constructions:

• The Brown–Loday non-abelian tensor square is defined by explicit generators and relations
• Rocco's crossed-product group $$\nu(H) = \langle H \cup H^\psi \mid [h_1, h_2^\psi] = [h_1 h_2, (h_1 h_2)^\psi],\ [h,h^\psi]=1\ \forall h, h_1, h_2 \in H \rangle$$
• The subgroup $\tau(H) = [H, H^\psi] < \nu(H)$ is isomorphic to $H \otimes H$
• There is a quotient map $\chi(H) \twoheadrightarrow \nu(H)$ with kernel $R(H)$

The consequence is that if $H$ is polycyclic-by-finite, so are $\nu(H)$ and $H \otimes H$, extending prior results for purely polycyclic $H$ to these broader classes.

4. Subgroup and Homological Structure

The subgroup landscape for $\chi(H)$ is intricate:

• $D(H)$, $L(H)$, $W(H)$, $R(H)$ form a lattice of normal subgroups
• The quotient $W(H)/R(H)$ is abelian and, for polycyclic-by-finite $H$, finitely generated (as is $L(H)' = \ker \varepsilon$)
• These subgroups relate to classic homological invariants, such as the Schur multiplier, and underpin the extension structure $$\chi(H) \twoheadrightarrow T(H) \twoheadrightarrow H$$, with $T(H)$ appropriately determined by $L(H)$ and $W(H)$

This architecture enables the transference of structure theorems dependent upon the preservation of finiteness or polynomial-type constraints in group cohomology, crucial for understanding non-abelian tensor products and the construction of central extensions.

5. Implications for Computational and Algorithmic Group Theory

The $\chi$ construction's compatibility with polycyclic and polycyclic-by-finite groups is significant for computational group theory:

• Polycyclic-by-finite groups admit effective polycyclic presentations with concrete algorithmic descriptions for group operations.
• Implementations (e.g., in GAP) can thus manipulate presentations of $\chi(H)$ and $H \otimes H$ when $H$ is polycyclic, enabling effective computation of relations, normal subgroups, and extension data.
• The preservation of finiteness and polycyclic-by-finite structure ensures that various algorithmic procedures, including those for isomorphism, membership, and coset enumeration, remain feasible in the weakly commuting context.

6. Broader Significance and Research Directions

The weak commutativity operator $\chi$ can be viewed as a universal diagonal embedding of $H$ into an enlargement enforcing only the minimal commutation relations $[h,h^\psi]=1$ for all $h \in H$. The identification of conditions under which $\chi(H)$, $H \otimes H$, and their various extensions remain in preferred group-theoretic classes (finite, solvable, nilpotent, polycyclic-by-finite) influences the paper of

• The interplay between group presentations, homological invariants (Schur multipliers), and crossed extensions
• The generalization of functorial constructions in the context of group cohomology and non-abelian homological algebra
• The explicit computational realization of non-abelian tensor squares and their applications in representation theory and related fields

The results in this context complete the “preservation under weak commutativity” paradigm by adding polycyclic and polycyclic-by-finite properties to the list, consolidating the status of $\chi(H)$ and associated constructions as computationally tractable and structurally robust for polycyclic inputs (Lima et al., 2014).