HNN Extension of a Free Group

• HNN Extension of a Free Group is a key construction in group theory that amalgamates isomorphic subgroups using stable letters, establishing a framework for Bass–Serre theory.
• The construction employs Britton's Lemma to ensure unique reduced normal forms, which is central to efficiently solving word and conjugacy problems in these groups.
• This topic underscores rich interconnections between algebraic structure, large-scale geometric properties such as Hilbert space compression, and algorithmic decision procedures.

An HNN extension of a free group is a fundamental construction in geometric and combinatorial group theory, encoding the amalgamation of isomorphic subgroups into a single group structure by adjoining stable letters that conjugate elements between these subgroups. This construction supports deep connections between algebraic, geometric, analytic, and algorithmic properties and is central to Bass–Serre theory, the paper of group actions on trees, and the analysis of mapping tori and automorphism dynamics. For free groups, HNN extensions serve as a testbed for exploring phenomena such as large-scale geometry, subgroup separability, residual properties, relative hyperbolicity, and decision problems.

1. Definitions, Structure, and Canonical Forms

Let $F$ be a finitely generated free group and let $A,B<F$ be isomorphic subgroups via $\phi:A\to B$. The associated HNN extension is

$$G = \operatorname{HNN}(F, A, B, \phi) = \langle F, t \mid t^{-1} a t = \phi(a), \; a\in A \rangle,$$

where $t$ is the stable letter.

Elements of $G$ admit unique reduced normal forms by Britton's Lemma: $$g = g_0 t^{\epsilon_1} g_1 t^{\epsilon_2} \cdots t^{\epsilon_k} g_k$$ with $g_i\in F$, $\epsilon_i = \pm1$, such that no subword $t^{-1} a t$ (with $a\in A$) or $t b t^{-1}$ (with $b\in B$) arises as a factor. This rigidity is the basis for algorithmic reductions and complexity bounds for the word problem in many classes of HNN extensions, especially when $A$ and $B$ are finite index or satisfy strong regularity constraints (Shen et al., 4 Oct 2025).

Multiple and basis-conjugating HNN extensions arise by introducing several stable letters $\{x_i\}$, each conjugating given basis elements by fixed words, and admit confluent and Noetherian normal forms via appropriate rewriting systems (Ionin, 23 Sep 2025). The interplay between Britton normal form and combinatorial diagrammatic methods underpins efficient algorithmic approaches to group calculations.

2. Geometric and Analytic Properties

Hilbert Space Compression

For an HNN extension $G = \operatorname{HNN}(H, F, \theta)$ with $F$ and $\theta(F)$ finite index in $H$ (with $H$ finitely generated), the non-equivariant Hilbert space compression $\alpha$ of $G$ is controlled in terms of $H$'s compression $\alpha_1$: $\frac{\alpha_1}{3} \leq \alpha \leq \alpha_1,$ and, in the equivariant setting, if $F\neq H$, one has

$\alpha = \min(\alpha_1, 1/2)$

(Dreesen, 2010). For free groups, this yields:

• Non-equivariant compression: $1/3 \leq \alpha \leq 1$
• Equivariant compression: $\alpha = 1/2$ (as for non-amenable groups in general)

These facts reflect how "complexity" added by the HNN extension distorts large-scale geometry, but within precisely quantifiable bounds, leveraging the action of $G$ on the Bass–Serre tree.

Metric Extension and Graev Metrics

Given a group $G$ with a two-sided invariant metric $d$, and isometric isomorphism $\phi:A\to B$ between closed subgroups, the HNN extension $$H = \langle G, t \mid t a t^{-1} = \phi(a),\, a\in A \rangle$$ can be equipped with a Graev-type two-sided invariant metric extending $d$ on $G$, provided $\operatorname{diam}(A)\le K$ (for some $K>0$), $d(t,e)=K$, and $\phi$ is an isometry (Slutsky, 2011). The explicit metric is constructed via a word metric minimizing over pairwise matched decompositions, analogously to free group metrics over pointed metric spaces.

Asymptotic Cones and Large-Scale Topology

Iterated HNN extensions of free groups over free associated subgroups exhibit a topological dichotomy in their asymptotic cones: every cone is either simply connected or has uncountable fundamental group, with no intermediate cases (Kent, 2012). This dichotomy emerges from van Kampen diagram techniques, specifically controlling t-annuli partitions and their translates, with the existence of nontrivial essential loops leading to uncountable free subgroups in the fundamental group of the cone.

Relative Hyperbolicity

For an ascending HNN extension $$F *_\phi = \langle F, t \mid t^{-1} f t = \phi(f) \ \forall f\in F \rangle$$ with $F$ finitely generated free and $\phi$ injective, non-surjective, and exponentially growing, $F *_\phi$ is hyperbolic relative to a canonical set of parabolic subgroups, specifically ascending HNN extensions over maximal polynomially growing subgroups (Krishna, 11 Dec 2024). This property fails when $\phi$ is not exponentially growing or lacks the strictly type-preserving condition (with respect to a free factor system).

3. Residual and Separability Properties

Order Separability

If $H = \langle F, t; t^{-1} a t = b\rangle$ is an HNN extension of a free group $F$ with cyclic subgroups $\langle a\rangle, \langle b\rangle$ maximal, then $H$ is 2-order separable: for any non-conjugate $u, v \in H$ (not conjugate to each other's inverses), there is a finite quotient $Q$ distinguishing their orders: $$\exists\,\phi:H \to Q,\quad |\phi(u)| \ne |\phi(v)|$$ (Yedynak, 2010). The property is a refinement over residual finiteness, providing strong control over finite quotients and supporting the construction of separating homomorphisms via action graph techniques. These results are significant for the richness of subgroup separability phenomena in HNN extensions of free groups.

4. Algorithmic Problems and Decision Procedures

Word Problem in Finite Index Case

For $G=F\ast_\varphi t$ with $F$ a free group, and $H_1=H_2=H<F$ equal associated subgroups of finite index (normal in $F$), the word problem in $G$ is solvable in polynomial time (Shen et al., 4 Oct 2025). The main ingredients are:

• Representation of long words via straight-line programs (SLPs) to avoid exponential blowup
• Britton's Lemma-based reduction, using subgroup graphs for $H$, spanning trees, and associated shift operations to implement compression-preserving syllable reductions
• For non-normal $H$, a normalizability condition on $\varphi$ allows reduction to the normal case within a finite-index subgroup

Conjugacy Problem for Ascending HNN Extensions

For $$F*_\phi = \langle F, t \mid t^{-1} ft = \phi(f)\rangle$$ with $F$ free, an algorithm is constructed for the conjugacy problem (Logan, 2022), using a reduction to the algorithmic solution of the $\phi^n$-twisted conjugacy problem in $F$, together with dynamic algorithms for fixed points of (possibly injective, non-surjective) endomorphisms and orbit enumeration in the associated pullback graphs. In particular, conjugacy in the HNN extension is equivalent to existence of $p,q$ and $x$ with $\phi^p(u) = x^{-1}\phi^q(v)\phi^n(x)$.

Submonoid Membership and One-Relator Inverse Monoids

Membership in certain submonoids of HNN extensions of free groups (with isomorphisms acting as bijections on subsets of a basis) is decidable precisely when the submonoid $Q\subset H$ satisfies $\phi(Q\cap A)=Q\cap B$ (with $A,B$ the associated subgroups). This decidability transfers to the prefix membership problem in one-relator groups—leading, via connections to inverse monoids, to new decidability results for word problems in one-relator inverse monoids (Warne, 7 Feb 2025). The decision procedure utilizes a finite set of elementary operations (addition, cancellation, semicommutation) and a computable "most reduced form" for HNN reduced words.

5. Subgroup Structures and Ping-Pong Lemmas

Constructive normal forms for multiple HNN extensions of a free group via basis-conjugating embeddings enable explicit verification of ping-pong conditions for collections of subgroups (Ionin, 23 Sep 2025). If, in such a group $G$, a family $\{A_i\}$ of subgroups is supported on disjoint subsets $T_i$ of stable letters (and each $A_i$ intersects the base free group trivially), then the natural free product $A_1*\dots*A_k\to G$ is injective. The proof employs a confluent rewriting system and analysis of the algebraic support of each word in its normal form to isolate ping-pong domains, with significant applications to the structure of the pure braid group and explicit construction of free subgroups in semidirect decompositions.

6. Analytic, Topological, and Measured Equivalence Aspects

$L^2$-Torsion, Thurston Norms, and BNS Invariant

For descending HNN extensions of free groups $G=F_n*_g$, Friedl–Lück's universal $L^2$-torsion $(2)_u(G)$ and its image under the polytope homomorphism $P_{L^2}(G)$ yield a Thurston seminorm on $H^1(G;\mathbb{R})$ generalizing that of $3$-manifold groups (Funke et al., 2016). The Thurston seminorm bounds all Alexander semi-norms, and in polynomially growing automorphic cases (unipotent on abelianization), these norms coincide. The Newton polytope of $(2)_u(G)$ locally determines the Bieri–Neumann–Strebel (BNS) invariant in rank two, with the invariant always possessing finitely many connected components. This provides a precise analytic and geometric structure for the first cohomology of such groups, driven by Fox calculus and algebraic $K$-theory.

Treeings and Measure Equivalence

For certain HNN extensions (notably Baumslag–Solitar groups), treeings can be constructed on quotient groupoids associated with measure-preserving actions, leading to the conclusion that the kernel of the modular homomorphism is measure equivalent to $F_\infty\times \mathbb{Z}$ (Kida, 2023). The construction leverages Bass–Serre theory to produce an explicit treeing in almost every fiber, linking measured group theory, cost invariants, and structural analysis of HNN extensions' modular subgroups.

7. Limitations, Pathologies, and Wild Phenomena

Infinite Palindromic Width

HNN extensions with proper associated subgroups (and amalgamated free products under mild index conditions) have infinite palindromic width: there is no uniform bound for the minimal number of palindromic factors needed to write arbitrary elements (Gongopadhyay et al., 2016). The proof utilizes quasi-homomorphisms tracking signature sequences in Britton normal forms and demonstrates the algebraic complexity inherent in these constructions. Notably, this property persists for most nontrivial graphs of groups, including those built from free groups.

Failure of the Howson and Finitely Generated Intersection Properties

Strictly ascending and descending HNN extensions of non-cyclic free groups systematically fail the Howson property: intersections of finitely generated subgroups may not be finitely generated (David, 2012, Bamberger et al., 2022). The underlying reason is the ability to stack infinitely many conjugates under the stable letter, resulting in wild, infinitely generated intersections. Sufficient dynamical conditions in relatively hyperbolic ascending HNN extensions guarantee FGIP failure, with applications to free-by-cyclic groups of exponential growth. For free groups themselves, the Howson property holds, but the extension via an injective non-surjective endomorphism instantly destroys it.

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These results encapsulate the rich algebraic, geometric, analytic, and computational structure of HNN extensions of free groups. They reveal how the addition of stable letters to free groups, even under strong index or dynamical constraints, dramatically alters group-theoretic, algorithmic, and geometric properties, while at the same time allowing strong quantitative control and explicit analytic or algorithmic techniques in special cases.