Higman--Thompson groups $F_n$ all the way down

Abstract: We prove that for every $n\ge 2$ the Higman--Thompson group $F_n$ has a maximal subgroup of infinite index isomorphic to itself. In fact, we construct a chain of subgroups $F_n=H_0>H_1>H_2>\cdots$, all isomorphic to $F_n$ and with trivial intersection, such that for every $i$ the only subgroups of $F_n$ containing $H_i$ are $H_i,H_{i-1},\ldots,H_0=F_n$; in particular, each $H_{i+1}$ is maximal in $H_i$. We prove that for all $n\ge m\ge 2$, every closed maximal subgroup of $F_m$ isomorphic to $F_n$ arises from a homeomorphism between the $n$-ary and $m$-ary Cantor spaces given by a finite semi-synchronizing transducer--a variation of the synchronizing transducers of Bleak, Cameron, Maissel, Navas and Olukoya. We characterize the homeomorphisms of Cantor spaces conjugating $F_n$ into $F_m$ as the order-preserving or order-reversing rational homeomorphisms whose minimal transducer is semi-synchronizing. At the heart of the paper is a machinery bridging transducers and Stallings $2$-cores of subgroups, which reduces the conjugation of finitely generated closed subgroups by such homeomorphisms to an algorithmic procedure. As applications, we prove that Jones' ternary oriented subgroup $\vec F_3\le F_3$ is isomorphic to $F_4$, answering questions of Aiello, and that all known maximal subgroups of infinite index of Thompson's group $F$ which act minimally on $(0,1)$ are isomorphic to Higman--Thompson groups. That raises the problem of whether all maximal subgroups of infinite index of $F$ which act minimally on $(0,1)$ are isomorphic to Higman--Thompson groups. We briefly discuss related results regarding fast groups of homeomorphisms and maximal subgroups of Thompson groups.

• The paper establishes that for each n ≥ 2, there exists a descending chain of maximal infinite index subgroups isomorphic to Fₙ, whose intersection is trivial.
• The methodology leverages transducer-induced automorphisms, semi-synchronizing transducers, and algorithmic tree automata to construct and classify the subgroup hierarchy.
• The findings reveal a rigid hierarchical structure of self-isomorphic subgroups, offering new insights into both the algebraic and dynamical properties of Higman–Thompson groups.

Descending Chains of Maximal Isomorphic Subgroups in Higman–Thompson Groups

Summary and Main Theorems

This paper investigates the subgroup structure of Higman–Thompson groups $F_n$, with a focus on the existence and nature of maximal infinite index subgroups isomorphic to $F_n$ itself. The principal result is that for each $n \geq 2$, there exists a descending chain of subgroups,

$F_n = H_0 > H_1 > H_2 > \cdots,$

where each $H_i$ is a maximal infinite index subgroup of $H_{i-1}$, all $H_i \cong F_n$, and $\bigcap_{i} H_i = \{1\}$. Furthermore, the chain is rigid in the sense that the only subgroups of $F_n$ containing $H_i$ are $F_n$0 and its predecessors in the chain.

The techniques introduced connect the theory of rational homeomorphisms of Cantor spaces (via transducers), the combinatorics of tree-diagram groups, and the machinery of core (tree-automaton) representations of closed subgroups. Additionally, the paper provides characterizations of all known maximal subgroups of infinite index of the Thompson group $F_n$1 and Higman–Thompson groups $F_n$2 acting minimally as being isomorphic to other Higman–Thompson groups.

Methods and Key Mechanisms

The main tools for the analysis and construction are:

• Transducer-induced automorphisms and rational homeomorphisms: Homeomorphisms between Cantor spaces $F_n$3 and $F_n$4 can be encoded by finite-state transducers, which in turn correspond to rational (finitely determined) homeomorphisms. The automorphism groups of generalized Thompson groups have been characterized via such transducers whose synchronizing properties are closely tied to the algebraic structure of the group (see [AutG], [AutT]).
• Semi-synchronizing transducers: The paper introduces and leverages the notion of semi-synchronization, a relaxation of the synchronizing condition appropriate for the endpoint-preserving context of $F_n$5. A homeomorphism $F_n$6 conjugating $F_n$7 into $F_n$8 is shown to be rational and semi-synchronizing, with the minimal transducer encoding critical dynamical and combinatorial invariants.
• Algorithmic manipulation of tree automata (cores): The core machinery developed provides an explicit and constructive means to identify closed subgroups, compute their core representations, and to algorithmically conjugate such subgroups via rational Cantor space homeomorphisms. The pullback and forward automata constructions precisely track the effect of conjugation on subgroup closure.
• Classification and rigidity of closed and maximal subgroups: By analyzing the possible tree-automaton quotients and generation conditions (adapted from the $F_n$9 case), the paper establishes that the only closed overgroups of a given $n \geq 2$0 are its chain predecessors, and that every properly containing subgroup must coincide with one of these.

Numerical and Structural Claims

• Rigidity and trivial intersection: Each $n \geq 2$1 is maximal of infinite index in $n \geq 2$2, the only intermediate closed subgroups are those along the chain, and their intersection is trivial.
• Isomorphism type of minimal, maximal subgroups: All known maximal infinite index subgroups acting minimally are isomorphic to Higman–Thompson groups $n \geq 2$3 for appropriate $n \geq 2$4.
• Explicit realization of Jones subgroups: E.g., Jones' oriented subgroup $n \geq 2$5 is shown to be isomorphic to $n \geq 2$6 via explicit construction of a transducer-conjugated copy.
• Algorithmic computability of conjugated cores: Given finitely generated closed $n \geq 2$7 and a rational semi-synchronizing conjugator $n \geq 2$8, the core for $n \geq 2$9 is explicitly constructible.

Theoretical Implications

The subgroup structure elucidated here demonstrates a rich rigidity: Higman–Thompson groups can contain infinitely many maximal infinite index copies of themselves in a controlled, hierarchical manner. This rigidity is witnessed both at the level of tree-diagram combinatorics and of the dynamical (transducer) realization of automorphisms. The conjunction of Cantor dynamics, transducer theory, and core automaton methods produces a powerful framework for understanding subgroup structure in groups of piecewise-linear homeomorphisms.

These results also strongly suggest that the phenomenon observed—the isomorphism of all minimal, maximal subgroups to some Higman–Thompson group—may be exhaustive. The natural open problem posed is whether every maximal infinite index subgroup of $F_n = H_0 > H_1 > H_2 > \cdots,$0 acting minimally is isomorphic to some $F_n = H_0 > H_1 > H_2 > \cdots,$1, and, more generally, to classify the types of such maximal subgroups for $F_n = H_0 > H_1 > H_2 > \cdots,$2.

Future Directions

Several avenues for further research arise:

• Classification of all minimal, maximal subgroups: Whether the observed pattern is universal, i.e., every such subgroup is isomorphic to some Higman–Thompson group, possibly subject to additional closedness or minimality conditions.
• Connections to diagram group theory and geometric generators: The techniques here intersect with the theory of diagram groups, in particular for groups generated by geometrically fast sets of one-bump functions, connecting subgroup isomorphism types to dynamical features of generating sets.
• Extensions to non-closed or non-minimal actions: The methods may be extended or adapted to address the structure of maximal subgroups outside the closed or minimal category, or to analogous subgroup rigidity in related (non-piecewise-linear) contexts.
• Algorithmic and computational group theory: The algorithmic reducibility of conjugation and closure for finitely generated subgroups in $F_n = H_0 > H_1 > H_2 > \cdots,$3 opens access to explicit computational analysis and, potentially, decidability results for membership or isomorphism problems.

Conclusion

The paper establishes a technically sophisticated and algorithmically constructive characterization of maximal infinite index subgroups in Higman–Thompson groups, revealing a recursive, rigid hierarchy of self-isomorphic subgroups and clarifying the automorphic and dynamical underpinnings of their structure. The interplay of Cantor dynamics, transducer automata, and the combinatorics of tree-diagram groups produces a unified view of subgroup rigidity in infinite simple groups of homeomorphisms, with clear implications for both algebraic and dynamical group theory.