Profinite Genus of HNN-Extensions

• Profinite genus of HNN-extensions is defined as the set of isomorphism classes of residually finite groups with identical finite quotients.
• Key findings reveal that infinite families of non-isomorphic HNN-extensions can share the same profinite completion, challenging traditional rigidity.
• Methodologies leverage Gaschütz’s lemma, Nielsen equivalence, and Bass–Serre theory to precisely enumerate orbits and distinguish complex group structures.

The profinite genus of HNN-extensions encapsulates the phenomenon of when group-theoretic invariants detectable in finite quotients fail to distinguish between non-isomorphic HNN-extensions. This area intersects group theory, topology, and the study of profinite completions, focusing on identifying, counting, and characterizing classes of HNN-extensions that are indistinguishable profinitely yet distinct as abstract groups.

1. Definitions and Fundamental Notions

An HNN-extension with base group $G_1$, associated subgroups $H, K \leq G_1$ and stable letter $t$ is defined as

$$\mathrm{HNN}(G_1, H, K, f) = \langle G_1, t \mid t h t^{-1} = f(h)\ \forall h \in H \rangle$$

where $f : H \to K$ is an isomorphism. The profinite completion $\widehat{G}$ of a group $G$ is the inverse limit of its finite quotient groups: $\widehat{G} = \varprojlim_{[G:H] < \infty} G/H$ The profinite genus $\g(G)$ of a finitely generated, residually finite group $G$ is the set of isomorphism classes of finitely generated residually finite groups $B$ such that $\widehat{B} \cong \widehat{G}$.

2. Profinite Completions and HNN-extensions

Given a residually finite $G_1$, one studies the profinite HNN-extension

$$\mathrm{HNN}(\widehat{G}_1, \widehat{H}, \widehat{K}, t, \widehat{f})$$

where $f$ extends continuously to the closures in the profinite topology. The key property is that in many natural cases, profinite completions of HNN-extensions are completely determined by their finite quotients, but this behavior admits both finiteness and infinite genus phenomena depending on the underlying structural features of the input data (Piwek, 2023, Bessa et al., 11 Jan 2026).

3. Infinite Profinite Genus among Free-by-Free and Related HNN-extensions

For certain kernels $N$, the construction detailed in (Piwek, 2023) produces, for each fixed $m \geq 2$ and subgroup $T < \mathrm{Aut}(N)$, infinite families of non-isomorphic groups

$G_i = N \rtimes_{\varphi_i} F_m$

with $N$ free of rank $\geq 10$, free abelian of rank $\geq 12$, or a surface group of genus $\geq 5$. Each $G_i$ shares the same profinite completion, i.e.,

$$\widehat{G}_i \cong \widehat{G}_j \qquad \forall i,j$$

but are pairwise non-isomorphic as abstract groups. The key mechanism is the selection of infinitely many pairwise non-T-equivalent surjections $\varphi_i : F_m \twoheadrightarrow T$ whose induced actions on $N$ cannot be interconverted by automorphisms of $F_m$ or $T$.

This stands in stark contrast to rigid cases, such as virtually polycyclic or virtually free groups, where the profinite genus is always finite and sometimes one (Piwek, 2023).

4. Profinite Genus for HNN-extensions with Finite Associated Subgroups

For HNN-extensions $\mathrm{HNN}(G_1, H, K, t, f)$ with $H, K$ finite, the profinite genus is governed by orbit counting in the space of possible gluing isomorphisms under the action of a group encapsulating both inner automorphisms and automorphisms preserving the subgroups: $$\#\{\text{isomorphism classes}\} = |\overline{\Gamma}_{HK} \backslash \mathrm{Iso}(H, K)|$$ where $\overline{\Gamma}_{HK}$ arises from the action of $G_1 \rtimes \mathrm{Aut}_{G_1}(H)$ with a possible added involution if $H$ and $K$ are interchanged by automorphisms of $G_1$ (Bessa et al., 11 Jan 2026).

In the profinite setting, the number of isomorphism classes of profinite HNN-extensions is bounded by the number of such orbits for the closures in the profinite topology. For normal HNNs (i.e., $H$ and $K$ conjugate), the profinite genus classifies to double cosets in $\mathrm{Out}(H)$ adjusted by the normalizers and automorphisms.

5. Methodologies for Computing and Distinguishing Profinite Genus

The construction of infinite profinite genus leverages several technical components:

• Gaschütz’s lemma (Profinite Schur–Zassenhaus), guaranteeing lifts of generating tuples through finite quotients.
• Application of Nielsen equivalence and T-equivalence on homomorphisms from free groups to the automorphism group $T$.
• Bass–Serre theory and its profinite analogue: the tree structure underlying abstract and profinite HNN-extensions is used to read off normalizers and invariant sets.
• Counting orbits under finite group actions for the precise enumeration of profinite genus in the finite subgroup case (Bessa et al., 11 Jan 2026).

Isomorphism distinctions between $G_i$ and $G_j$ for $i \neq j$ are typically established by showing that any hypothetical isomorphism must intertwine the inducing automorphisms up to outer automorphisms in a way that is explicitly precluded by the construction (Piwek, 2023).

6. Explicit Formulas and Special Cases

Some explicit formulas and results include:

• For cyclic associated subgroups $H \cong C_n$, the profinite genus

$|\g| = \begin{cases} 1 & n \leq 2 \ \frac{\phi(n)}{2} & n \geq 3 \end{cases}$

where $\phi$ is Euler's totient function (Bessa et al., 11 Jan 2026).

• If the base group $G_1$ is finitely generated abelian and $H \neq K$, or if $H$ is center or malnormal in $G_1$, or if the relevant normalizers are trivial or small, the profinite genus collapses to one.
• For Fuchsian base groups or surface groups with finite cyclic $H$, the profinite genus is always one, as all automorphisms lift (Bessa et al., 11 Jan 2026).

7. Significance and Broader Implications

The existence of infinite profinite genus in non-abelian HNN-extensions with free or surface kernels marks a fundamental failure of profinite rigidity in a new, substantial class of groups. No finite suite of invariants surviving profinite completion can distinguish these groups. For many classical families (such as virtually polycyclic, virtually free, or 3-manifold groups), the profinite genus is always finite and often rigid, but these new examples demonstrate the inherent limitations of profinite techniques in distinguishing abstract group structures (Piwek, 2023).

Plausible implications include a reevaluation of the scope of profinite rigidity for large classes of groups and renewed interest in constructing invariants distinguishable beyond the field of finite quotients. These results are also consequential for understanding the limits of Galois rigidity phenomena in arithmetic and geometric topology contexts.