On $\textbf{u}$-substitutions for group presentations

Abstract: We investigate groups $\hat{G}$ given by a presentation $\mathcal{P}=\langle \textbf{x}: \textbf{r} \rangle$ whose relators $\textbf{r} \subseteq F(\textbf{x})$ are comprised of a set of subwords in $F(\textbf{x})$, i.e. $\textbf{r}$ admits a $\textbf{u}$-substitution in the sense that there exists a homomorphsim $ε: F(\textbf{u}) \rightarrow F(\textbf{x})$ and a subset $\mathbf{v} \subseteq F(\textbf{u})$ such that $\textbf{r}=ε(\textbf{v})$. Equivalently, $$\mathcal{P}= \langle \textbf{x}: ε(\textbf{v}) \rangle$$ is referred to as the composition of the presentation $$\mathcal{G}= \langle \textbf{u}: \textbf{v} \rangle$$ with $$\mathcal{H}= \langle \textbf{x}: ε(\textbf{u}) \rangle$$ of the groups $G$ and $H$, respectively. We survey known results and record structural properties which do not explicitly appear in the literature, e.g. that there is the relative presentation $$\langle G, \textbf{x}: \textbf{u}= ε(\textbf{u}) \rangle$$ for $\hat{G}$. Thus there is a natural map $ε: G \rightarrow \hat{G}$ and we may consider the associated problems (e.g. injectivity, finiteness). As an application, we investigate the class of groups $\mathcal{G}(\mathcal{B})$ obtained from substituting a presentation of the trivial group into a deficiency one group presentation for $\mathbb{Z}$. Our results show $\mathcal{G}(\mathcal{B})$ is a proper subset of the class of 2-knot groups and properly contains all classical knot groups. A subfamily is investigated which includes the group of the trefoil knot and uses Higman's presentations for the trivial group.

• The paper develops a relative-presentation framework for substitutions and derives the exact sequence 1 → ⟨⟨ε(G)⟩⟩ → Ĝ → H → 1, linking injectivity to equations over groups and asphericity.
• The paper proves that relative asphericity ensures injectivity and constrains finite subgroups, while examples show that composite groups can vary substantially and may be finite or infinite under similar substitutions.
• The paper establishes the strict inclusion K₁ ⊊ 𝒢(ℬ) ⊊ K₂ for 2-knot groups, includes all deficiency-one Wirtinger presentations, and identifies open problems involving ribbon knots, Q(3,1), and Andrews–Curtis equivalence.

The construction and its exact sequence

A word $w \in F(\mathbf{x})$ is composite if it can be assembled from a set of subwords via a substitution homomorphism $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$, so that a presentation $$\mathcal{P} = \langle \mathbf{x}: \mathbf{r} \rangle$$ with $\mathbf{r} = \epsilon(\mathbf{v})$ is the composition of $$\mathcal{G} = \langle \mathbf{u}: \mathbf{v} \rangle$$ (presenting $G$) with $$\mathcal{H} = \langle \mathbf{x}: \epsilon(\mathbf{u}) \rangle$$ (presenting $H$). Although composite words pervade combinatorial group theory—Nielsen transformations, Magnus's work on one-relator presentations, Fox's free differential calculus—the paper observes that no systematic treatment of composite group presentations exists, and sets out to develop one.

The central structural result is that the substitution induces a natural map $\epsilon: G \rightarrow \hat{G}$ fitting into an exact sequence

$$1 \rightarrow \langle\langle \epsilon(G) \rangle\rangle \rightarrow \hat{G} \xrightarrow{\pi} H \rightarrow 1.$$

The proof adjoins the generators $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$0 with defining relations $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$1 via Tietze transformations, yielding the relative presentation

$\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$2

which frames injectivity of $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$3 as an instance of the adjunction problem from equations over groups. This relative presentation, and the associated pair of cellular models $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$4, are stated as not appearing explicitly in prior literature. When both $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$5 and $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$6 are aspherical, the composite presentation $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$7 is aspherical; moreover, relative asphericity implies $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$8 is injective, confines finite subgroups of $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$9 to conjugates of subgroups of $$\mathcal{P} = \langle \mathbf{x}: \mathbf{r} \rangle$$0, and forces $$\mathcal{P} = \langle \mathbf{x}: \mathbf{r} \rangle$$1 infinite whenever both $$\mathcal{P} = \langle \mathbf{x}: \mathbf{r} \rangle$$2 and $$\mathcal{P} = \langle \mathbf{x}: \mathbf{r} \rangle$$3 are nontrivial.

The normal closure of $$\mathcal{P} = \langle \mathbf{x}: \mathbf{r} \rangle$$4 and finiteness

The gap between $$\mathcal{P} = \langle \mathbf{x}: \mathbf{r} \rangle$$5 and its normal closure in $$\mathcal{P} = \langle \mathbf{x}: \mathbf{r} \rangle$$6 is quantified in free-group terms: $$\mathcal{P} = \langle \mathbf{x}: \mathbf{r} \rangle$$7 while $$\mathcal{P} = \langle \mathbf{x}: \mathbf{r} \rangle$$8, so the index of $$\mathcal{P} = \langle \mathbf{x}: \mathbf{r} \rangle$$9 in its normal closure equals an index computable in $\mathbf{r} = \epsilon(\mathbf{v})$0. Two explicit finite examples show this index can be one: $\mathbf{r} = \epsilon(\mathbf{v})$1 extending $\mathbf{r} = \epsilon(\mathbf{v})$2 by $\mathbf{r} = \epsilon(\mathbf{v})$3, and dihedral groups $\mathbf{r} = \epsilon(\mathbf{v})$4 of order $\mathbf{r} = \epsilon(\mathbf{v})$5 arising from $\mathbf{r} = \epsilon(\mathbf{v})$6 with $\mathbf{r} = \epsilon(\mathbf{v})$7. Beyond such cases, however, the paper reports that GAP experimentation suggests $\mathbf{r} = \epsilon(\mathbf{v})$8 is typically infinite even when $\mathbf{r} = \epsilon(\mathbf{v})$9 is finite cyclic and $$\mathcal{G} = \langle \mathbf{u}: \mathbf{v} \rangle$$0 trivial—a small census of finite examples was found, all with $$\mathcal{G} = \langle \mathbf{u}: \mathbf{v} \rangle$$1 or odd $$\mathcal{G} = \langle \mathbf{u}: \mathbf{v} \rangle$$2, and the search is explicitly non-exhaustive and limited by relator length growth ($$\mathcal{G} = \langle \mathbf{u}: \mathbf{v} \rangle$$3).

Two consequences deserve emphasis. First, the finite examples show $$\mathcal{G} = \langle \mathbf{u}: \mathbf{v} \rangle$$4 is not an invariant of $$\mathcal{G} = \langle \mathbf{u}: \mathbf{v} \rangle$$5 nor of $$\mathcal{G} = \langle \mathbf{u}: \mathbf{v} \rangle$$6: two substitutions on $$\mathcal{G} = \langle \mathbf{u}: \mathbf{v} \rangle$$7 yield non-isomorphic groups $$\mathcal{G} = \langle \mathbf{u}: \mathbf{v} \rangle$$8 and $$\mathcal{G} = \langle \mathbf{u}: \mathbf{v} \rangle$$9. Second, iterated substitutions produce endomorphisms $G$0 that fail to be surjective yet for which $G$1 is finite cyclic of order $G$2 (or $G$3 in odd rank), giving proper descending chains of finite-index normal subgroups normally generated by the images of the generators—examples where every $G$4 has word length greater than one and exponential growth under iteration.

Knot groups between $G$5 and $G$6

Composing a deficiency-one presentation of $G$7 with a balanced presentation of the trivial group yields the class $G$8: finitely presented groups of weight one, deficiency one, and infinite cyclic abelianization—hence 2-knot groups by the Kervaire conditions. The main containment theorem states

$G$9

with properness on the left witnessed by the group $$\mathcal{H} = \langle \mathbf{x}: \epsilon(\mathbf{u}) \rangle$$0, whose Alexander polynomial $$\mathcal{H} = \langle \mathbf{x}: \epsilon(\mathbf{u}) \rangle$$1 is not symmetric; properness on the right follows from Kervaire's existence of deficiency-zero 2-knot groups. Every deficiency-one Wirtinger presentation admits such a substitution, so the class also contains all ribbon 2-knot groups. Whether that containment is proper is left open, and is tied to Neumann's potential Andrews–Curtis counterexample: if Neumann's presentation is not AC-equivalent to the trivial presentation, then $$\mathcal{H} = \langle \mathbf{x}: \epsilon(\mathbf{u}) \rangle$$2 strictly contains the ribbon 2-knot groups.

Injectivity, surjectivity, and equations over groups

Injectivity of $$\mathcal{H} = \langle \mathbf{x}: \epsilon(\mathbf{u}) \rangle$$3 holds when the substitution images partition the generators (iterated amalgamated products), when the relative presentation is aspherical, and—for proper powers $$\mathcal{H} = \langle \mathbf{x}: \epsilon(\mathbf{u}) \rangle$$4 with nonzero exponent sum—by Rothaus's theorem applied to the resulting independent system of equations over the residually finite coefficient group $$\mathcal{H} = \langle \mathbf{x}: \epsilon(\mathbf{u}) \rangle$$5. Injectivity fails in general: the substitution $$\mathcal{H} = \langle \mathbf{x}: \epsilon(\mathbf{u}) \rangle$$6 kills the product $$\mathcal{H} = \langle \mathbf{x}: \epsilon(\mathbf{u}) \rangle$$7, and the corresponding shift-extension equation $$\mathcal{H} = \langle \mathbf{x}: \epsilon(\mathbf{u}) \rangle$$8 is singular precisely because torsion in $$\mathcal{H} = \langle \mathbf{x}: \epsilon(\mathbf{u}) \rangle$$9 obstructs solvability. Notably, a composite $H$0 need not be residually finite even when $H$1 is injective and both $H$2 and $H$3 are residually finite, via the Baumslag–Miller–Troeger example. Conversely, every cyclically presented group embeds as a subgroup that normally generates the kernel of some substitution into an SQ-universal cyclically presented group (for $H$4), using Levin's theorem on equations with positive exponents. Surjectivity of the induced map is characterized as equivalent to $H$5 together with normality of $H$6, and is algorithmically undecidable since it subsumes the triviality problem.

The family $H$7

The paper studies $H$8, composing $H$9 (cyclic of order 2 or infinite) with modifications of Higman's groups $\epsilon: G \rightarrow \hat{G}$0, which are trivial for $\epsilon: G \rightarrow \hat{G}$1 and SQ-universal for $\epsilon: G \rightarrow \hat{G}$2. For $\epsilon: G \rightarrow \hat{G}$3, $\epsilon: G \rightarrow \hat{G}$4 is normally generated by the involution-related element $\epsilon: G \rightarrow \hat{G}$5; it lies in $\epsilon: G \rightarrow \hat{G}$6 when $\epsilon: G \rightarrow \hat{G}$7 is even, and is otherwise finitely generated by involutions, being a quotient of a group in $\epsilon: G \rightarrow \hat{G}$8 by $\epsilon: G \rightarrow \hat{G}$9. All $$1 \rightarrow \langle\langle \epsilon(G) \rangle\rangle \rightarrow \hat{G} \xrightarrow{\pi} H \rightarrow 1.$$0 have trivial second homology, proved via Hopf's theorem and an explicit spherical diagram generating $$1 \rightarrow \langle\langle \epsilon(G) \rangle\rangle \rightarrow \hat{G} \xrightarrow{\pi} H \rightarrow 1.$$1 of the presentation complex. Key identifications include:

Group	Structure
$$1 \rightarrow \langle\langle \epsilon(G) \rangle\rangle \rightarrow \hat{G} \xrightarrow{\pi} H \rightarrow 1.$$2	trefoil knot group $$1 \rightarrow \langle\langle \epsilon(G) \rangle\rangle \rightarrow \hat{G} \xrightarrow{\pi} H \rightarrow 1.$$3
$$1 \rightarrow \langle\langle \epsilon(G) \rangle\rangle \rightarrow \hat{G} \xrightarrow{\pi} H \rightarrow 1.$$4	$$1 \rightarrow \langle\langle \epsilon(G) \rangle\rangle \rightarrow \hat{G} \xrightarrow{\pi} H \rightarrow 1.$$5, a 2-knot group
$$1 \rightarrow \langle\langle \epsilon(G) \rangle\rangle \rightarrow \hat{G} \xrightarrow{\pi} H \rightarrow 1.$$6, $$1 \rightarrow \langle\langle \epsilon(G) \rangle\rangle \rightarrow \hat{G} \xrightarrow{\pi} H \rightarrow 1.$$7 odd	free products of copies of $$1 \rightarrow \langle\langle \epsilon(G) \rangle\rangle \rightarrow \hat{G} \xrightarrow{\pi} H \rightarrow 1.$$8 amalgamated over $$1 \rightarrow \langle\langle \epsilon(G) \rangle\rangle \rightarrow \hat{G} \xrightarrow{\pi} H \rightarrow 1.$$9
$\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$00	$\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$01; finiteness undetermined

Modulo the squares of generators, $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$02 decomposes as a free product of $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$03 dihedral groups $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$04, $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$05, amalgamated along order-2 subgroups. Both $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$06 and $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$07 share the quotient $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$08 modulo $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$09 and $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$10 modulo $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$11. The group $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$12 remains unresolved: it may be finite, though it surjects onto $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$13, and its quotient by $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$14 is described as a quotient of Newman's one-relator group $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$15, which is not residually torsion-free nilpotent.

Limitations and open questions

The paper concedes several boundaries of its results. The computational census of finite $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$16 is selective rather than exhaustive, constrained by relator-length growth, and no finite example with $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$17 beyond the two infinite families was found. Whether $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$18 properly contains the ribbon 2-knot groups is open and conditional on the Andrews–Curtis status of Neumann's presentation. The finiteness of $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$19 is undetermined. The author notes that asphericity failures of the relative presentation give rise to essential maps $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$20, representatives of nontrivial $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$21, whose analysis is deferred to forthcoming work. Finally, whether the triviality of $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$22 with nontrivial $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$23 can occur connects to the Kervaire conjecture; shift extensions always retract onto $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$24 and thus cannot supply counterexamples there, though the cyclically presented groups themselves remain candidates.

Conclusion

The paper organizes composite group presentations around a single exact sequence and its associated relative presentation, connecting the construction to adjunction problems, asphericity, and the Kervaire conditions. Its principal concrete contribution is the strict chain $\epsilon: F(\mathbf{u}) \rightarrow F(\mathbf{x})$25 of knot groups, instantiated by families built from Higman's presentations of the trivial group, together with explicit finite and infinite exemplars delimiting when the substitution map preserves finiteness, injectivity, or normality.