1 Introduction↩︎

Thompson’s group \(F\) was introduced by R. Thompson in the 1960s and has since become a central example in geometric and combinatorial group theory. It can be realized as the group of all piecewise-linear orientation-preserving homeomorphisms of the unit interval whose breakpoints are dyadic rationals and whose slopes are powers of \(2\). Brown studied the corresponding \(n\)-adic analogues \(F_n\), \(n\ge 2\), which are known as Higman–Thompson groups: \(F_n\) consists of the piecewise-linear homeomorphisms of \([0,1]\) with breakpoints in \(\mathbb{Z}[\frac{1}{n}]\) and slopes powers of \(n\); thus \(F=F_2\) [@B1]. These groups also admit a combinatorial description by pairs of finite \(n\)-ary trees with the same number of leaves, as well as a description as diagram groups in the sense of Guba and Sapir [@GubaSapir1997].

The present paper studies the generation problem for \(F_n\). For a finitely generated group \(G\), the generation problem asks whether there is an algorithm which, given a finite subset \(X\subseteq G\), decides whether \(X\) generates \(G\). This problem is undecidable in some finitely generated groups, for instance in the direct product of a rank-two free group with itself; see, for example, [@LyndonSchupp1977]. For Thompson’s group \(F=F_2\), the first author solved the generation problem in [@G1] using the Stallings \(2\)-core of a subgroup. Our first goal is to extend the relevant core and semi-core methods to the Higman–Thompson groups \(F_n\) and to obtain effective sufficient conditions for generation.

The core of a subgroup of a diagram group was introduced by Guba and Sapir. For subgroups of Thompson’s group \(F\), the construction adapts naturally to the language of tree diagrams and automata [@GS1; @G2]. We use the analogous construction for \(F_n\). If \(H\le F_n\), its core, denoted \(\mathcal{L}(H)\), is a rooted edge-labeled automaton constructed from tree diagrams for a generating set of \(H\) by identifying the corresponding roots and leaves and then applying two types of foldings. Such an automaton accepts a tree diagram when the branches of the two trees can be read from the root and paired branches end at the same vertices. An element of \(F_n\) is accepted if it has an accepted tree-diagram representative. The set of all elements accepted by \(\mathcal{L}(H)\) is the closure of \(H\), denoted \(\mathrm{Cl}(H)\). This operation satisfies the usual properties of a closure operator: \(H\le \mathrm{Cl}(H)\), \(\mathrm{Cl}(\mathrm{Cl}(H))=\mathrm{Cl}(H)\), and \(H_1\le H_2\) implies \(\mathrm{Cl}(H_1)\le \mathrm{Cl}(H_2)\).

A related object, the semi-core, was introduced by the first author for subsets of \(F\) [@G1]. We adapt the construction to \(F_n\). If \(X\subseteq F_n\), the semi-core \(\mathcal{L}_{sem}(X)\) is obtained from the tree diagrams of the elements of \(X\) by identifying the corresponding roots and leaves and then applying only foldings of type \(1\). Unlike the core, the semi-core depends on the chosen generating set rather than only on the subgroup it generates. This dependence is useful: identifications in \(\mathcal{L}_{sem}(X)\) can be used to produce explicit elements of the subgroup \(\langle X\rangle\) with prescribed local behavior on \(n\)-adic intervals.

We shall use one simple invariant. For a finite \(n\)-ary word \(w\), let \(t(w)\) be the sum of the digits of \(w\) modulo \(n-1\). Every point \(\alpha\in \mathbb{Z}[\frac{1}{n}]\cap(0,1)\) has a finite \(n\)-ary expansion, unique up to appending zeros, and we define \(t(\alpha)\) to be the sum of the digits in such an expansion modulo \(n-1\). For \(i\in\{0,\dots,n-2\}\), let \[D_{n,i}=\{\alpha\in \mathbb{Z}[\tfrac1n]\cap(0,1)\mid t(\alpha)=i\}.\] The sets \(D_{n,0},\dots,D_{n,n-2}\) are precisely the orbits of the action of \(F_n\) on the \(n\)-adic points in \((0,1)\). Thus, for \(F_2\) there is only one such orbit, while for \(F_n\) with \(n>2\) this congruence invariant is one of the main new features.

Our main generation criterion is the following theorem. Here, if a word \(w\) labels a path in an automaton, \(w^+\) denotes the terminal vertex of that path.

The first two conditions separate the problem into the part detected by the closure and the part detected by the abelianization. Condition \((1)\) says that the closure of \(H\) contains the derived subgroup of \(F_n\); for finitely generated \(H\), this can be checked on the finite core by verifying that every vertex is a father and that there are exactly \(n-1\) inner vertices. Condition \((2)\) says that the image of \(H\) in the abelianization of \(F_n\) is the whole abelianization. Condition \((3)\) supplies, in each congruence class \(D_{n,i}\), an element of \(H\) with a prescribed one-sided slope behavior at a fixed point. We give a tuple algorithm which decides this condition for finitely generated subgroups satisfying \((1)\).

Condition \((4)\) is the semi-core condition. It says that, after reading a fixed inner prefix \(u\), the relevant rooted part of the semi-core has the same inner structure as the core of \(F_n\): among paths starting with \(u\), the terminal vertex depends only on the value of \(t\). Under condition \((1)\), the corresponding identifications already hold in the core \(\mathcal{L}(H)\). The semi-core condition is used to ensure that these identifications can be realized by elements of the subgroup \(H=\langle X\rangle\) itself, rather than only by elements of its closure.

For Thompson’s group \(F=F_2\), this extra condition is superfluous. In [@G1], it was proved that the analogues of conditions \((1)\), \((2)\) and \((3)\) already imply that \(H=F\). This gives a necessary and sufficient generation criterion for \(F\) and solves the generation problem in \(F\). Moreover, in later work on maximal subgroups of \(F\) [@G2], it was proved that, for subgroups of \(F\), the analogues of conditions \((1)\) and \((2)\) already imply the analogue of condition \((3)\). Thus, for \(F\), generation is detected by the closure and the image in the abelianization alone. One of the natural questions left by the present paper is whether analogous simplifications hold for \(F_n\) when \(n>2\).

The second application of these methods concerns maximal subgroups. Savchuk initiated the study of maximal subgroups of infinite index in Thompson’s group \(F\) by proving that, for every \(\alpha\in(0,1)\), the stabilizer of \(\alpha\) in \(F\) is a maximal subgroup of infinite index, and he asked whether there are other examples [@Sav1; @Sav2]. In answer to this question, the first author and Mark Sapir constructed an explicit maximal subgroup of infinite index in \(F\) which does not fix any point of \((0,1)\) [@GS1]. This subgroup is the preimage of Jones’ oriented subgroup \(\vec{F}\) under an injective endomorphism of \(F\); recall that \(\vec{F}\) is isomorphic to \(F_3\).

The work of the first author and Mark Sapir also provided a general method for proving the existence of many other maximal subgroups of \(F\) [@GS1; @G2]. Aiello and Nagnibeda constructed three more explicit maximal subgroups of infinite index in \(F\), relying on the techniques from [@GS1]; see [@AN2]. In [@G2], the first author proved that \(F\) has infinitely many non-isomorphic maximal subgroups of infinite index by constructing, for each prime \(p\), a maximal subgroup of \(F\) isomorphic to \(F_{p+1}\).

The search for maximal subgroups naturally extends to the Higman–Thompson groups \(F_n\). Savchuk’s proof readily adapts to this setting, showing that for every \(\alpha\in(0,1)\), the stabilizer of \(\alpha\) in \(F_n\) is a maximal subgroup of infinite index. Aiello and Nagnibeda provided the first example of a maximal subgroup of infinite index in \(F_3\) which does not fix a point of \((0,1)\) [@AN]. Their example is the preimage of the Jones subgroup of \(F_3\) under an injective endomorphism of \(F_3\).¹ They asked whether, for every \(n\ge2\), the group \(F_n\) contains a maximal subgroup of infinite index which does not fix any point of \((0,1)\).

We answer this question affirmatively for every \(n\ge2\).

The proof of Theorem 2 uses the generation criterion above together with a detailed analysis of core automata. We start from the tree semigroup presentation associated with the core of \(F_{2n-1}\). We then construct a sequence of semigroup presentations by applying basic splittings, and pass to the corresponding rooted semi tree automata. After that, we apply foldings of type \(2\). The splitting steps preserve the associated semigroup and, by a theorem of Guba and Sapir, the isomorphism type of the corresponding diagram group; the folding steps preserve the diagram group as well. This yields a core automaton for a subgroup \(H\le F_n\) with \(H\cong F_{2n-1}\). Theorem 1 is then used to show that adjoining any element of \(F_n\setminus H\) generates all of \(F_n\), and hence that \(H\) is maximal.

The paper is organized as follows. In Section 2, we recall the definitions of \(F_n\) as a group of piecewise-linear homeomorphisms and as a group of tree diagrams, and we review the core, closure, abelianization, and derived subgroup of \(F_n\). In Section [section:gen_problem], we prove the generation criterion and develop the tuple algorithm used to verify condition \((3)\) above. In Section 4, we study rooted tree automata and the operations on the corresponding semigroup presentations that preserve the associated diagram groups. In Section 5, we construct the maximal subgroup isomorphic to \(F_{2n-1}\) and prove Theorem 2. We conclude in Section 6 with some open problems, and with a brief discussion of subsequent related work.

Acknowledgements. The results proved in this paper formed part of the second author’s M.Sc. thesis, submitted at Ben-Gurion University of the Negev in September 2025 under the supervision of the first author. The research of both authors was partially supported by Israel Science Foundation grant Nos. 2322/19 and 2275/24.

2 Preliminaries↩︎

2.1 Preliminaries on \(F_n\)↩︎

We start this section by listing some definitions and notations that are relevant for the rest of this paper.

All paths in an \(n\)-ary tree in this paper will be rooted paths (that is, their initial vertex is the root), so we will usually omit the word “rooted”. A path ending in a leaf is called a branch. Each path \(p\) in an \(n\)-ary tree can be naturally labeled by an \(n\)-ary word \(lab(p)\). Explicitly, we label the path consisting only of the root by the empty string. Then, if \(p\) is a path, we label the paths extending \(p\) by a single edge as \(lab(p)i\) for \(i\in\{0,\dots,n-1\}\). Throughout this paper, we will often not distinguish between a rooted path, its label, and the vertex at the end of the path. For the end vertex \(v\) of a path labeled by a non-empty \(n\)-ary word \(\sigma\equiv\sigma_1\dots\sigma_l\), we denote \(.v:=.\sigma=\sum_{j=1}^l{\sigma_j\cdot n^{-j}}\) and \([v]:=[\sigma]=[.v,.v+\frac{1}{n^l}]\). Note that given an \(n\)-ary tree with leaves \(v_1,\dots,v_m\), the intervals \([v_1],\dots,[v_m]\) partition the unit interval into \(n\)-adic intervals, providing an \(n\)-adic subdivision.

An element of \(F_n\) can be viewed as an equivalence class of pairs of \(n\)-ary trees with the same number of leaves. Let \((T_1,T_2)\) be an \(n\)-ary tree-diagram. Let the (labels of the) branches of \(T_1\) be \(u_1,\dots,u_m\), and the branches of \(T_2\) be \(v_1,\dots,v_m\). For any \(1\le j\le m\), we say that \((T_1,T_2)\) has the pair of branches \(u_j\rightarrow v_j\). The tree-diagram corresponds to a function \(f \in F_n\) that maps each interval \([u_j]\) linearly onto the interval \([v_j]\).

A caret under \(w\) is a tree consisting of a root vertex \(w\) and \(n\) leaf nodes under \(w\) (often, we omit the reference to \(w\) and simply call it a caret). A tree-diagram \((T_1',T_2')\) is obtained from \((T_1,T_2)\) by adding a common caret if \(T'_1\) and \(T'_2\) are obtained by adding carets under leaves \(u\) and \(v\) respectively, where \(u\rightarrow v\) is a pair of branches of \((T_1,T_2)\). Elements of \(F_n\) are defined as equivalence classes of tree-diagrams, where two diagrams are equivalent (denoted by \(\sim\)) if one can pass from one to the other via a finite sequence of adding or removing common carets. If a tree-diagram cannot be obtained by adding a common caret to another tree-diagram it is called reduced; otherwise it is called reducible. Every tree-diagram has a unique equivalent reduced tree-diagram (see [@CFP1996] for example).

Note that for every two tree-diagrams \((T_1,T_2),(T_3,T_4)\), there exist (respectively) equivalent tree-diagrams \((T_1',T_2'),(T_3',T_4')\) such that \(T_2'=T_3'\). With this notation, the product of two equivalence classes of tree-diagrams \((T_1,T_2)\cdot (T_3,T_4)\) is the equivalence class of \((T_1',T_4')\). This construction provides an isomorphism between \(F_n\) as a group of homeomorphisms and its description as a group of equivalence classes of tree-diagrams (for more details about the point of view as tree-diagrams, see [@CFP1996] for the \(F_2\) case, [@B1] for the generalization to \(F_n\), and [@GubaSapir1997] for the broader context of diagram groups).

A well-known fact is that \(F_n\) acts transitively on \(D_{n,i}\) for all \(i\in\{0,\dots,n-2\}\) (see, for example, [@Stein]). We will later use the following slightly stronger claim.

More generally, one can show the following.

2.2 The core and closure of a subgroup of \(F_n\)↩︎

The core of a subgroup of a diagram group was originally defined in [@GS1] using the language of directed \(2\)-complexes. Later, it was described specifically for Thompson’s group \(F\) in the narrower language of tree-diagrams and tree-automata ([@G2]). We follow the latter approach and provide a definition of the core for subgroups of \(F_n\) using automata.

Recall that an automaton \(\mathcal{A}\) is a directed, edge-labeled graph. Every automaton considered in this paper has a distinguished vertex \(r\) called the root. We denote such an automaton by \(\mathcal{A}_r\) and refer to it as a rooted automaton. Throughout this paper, a path in a rooted automaton \(\mathcal{A}_r\) refers to a finite directed path starting from the root.

Let \(\mathcal{A}_r\) be an \(n\)-ary tree-automaton. A vertex in \(\mathcal{A}_r\) with no outgoing edges is called a leaf. Each path in \(\mathcal{A}_r\) can be labeled by an \(n\)-ary word \(\sigma\); we will usually not distinguish between a path and its label. Note that an \(n\)-ary word labels at most one path in \(\mathcal{A}_r\). If an \(n\)-ary word \(\sigma\) labels a path in \(\mathcal{A}_r\), we say that \(\sigma\) is readable on \(\mathcal{A}_r\), and we denote the end vertex of this path by \(\sigma^+\).

Similarly, given a vertex \(u\) in an \(n\)-ary tree \(T\), we say that the vertex \(u\) or that the path \(u\) is readable on \(\mathcal{A}_r\) if the word labeling the rooted path to \(u\) is readable on \(\mathcal{A}_r\); in this case, we denote the end vertex of this path in \(\mathcal{A}_r\) by \(u^+\). An \(n\)-ary tree \(T\) is said to be readable on \(\mathcal{A}_r\) if every leaf of \(T\) is readable on \(\mathcal{A}_r\).

The proof of this lemma for \(F=F_2\) was given in [@G2]. The proof for \(F_n\) is almost identical, so we omit it.

We say that an element \(f\in F_n\) is accepted by an \(n\)-ary tree-automaton \(\mathcal{A}_r\) if the reduced tree-diagram representing \(f\) is accepted by \(\mathcal{A}_r\). Equivalently, by Lemma 14, \(f\) is accepted by \(\mathcal{A}_r\) if there exists a tree-diagram representing \(f\) that is accepted by \(\mathcal{A}_r\). By Lemma 14, given an \(n\)-ary tree-automaton \(\mathcal{A}_r\), the set of all elements in \(F_n\) accepted by \(\mathcal{A}_r\) forms a subgroup of \(F_n\). Following [@G2] we make the following definitions.

Note that diagram groups over rooted tree-automata are a special case of the diagram groups studied by Guba and Sapir [@GubaSapir1997; @GubaSapir1999subgroups].

Let \(H\) be a subgroup of \(F_n\). We want to find the smallest closed subgroup of \(F_n\) that contains \(H\). To do so, we define a rooted \(n\)-ary tree-automaton called the core of \(H\).

Note that unlike the core, the semi-cores of different generating sets of a subgroup \(H\leq F_n\) might not be isomorphic. Note also that if \(H\leq F_n\), then every element in \(H\) is accepted by the core \(\mathcal{L}(H)\).

The closure operation satisfies standard properties of a closure operator. Namely, if \(H\) is a subgroup of \(F_n\), then \(H\le\mathrm{Cl}(H)\) and \(\mathrm{Cl}(\mathrm{Cl}(H))=\mathrm{Cl}(H)\); furthermore, if \(H_1\le H_2\) are subgroups of \(F_n\), then \(\mathrm{Cl}(H_1)\le\mathrm{Cl}(H_2)\).

Let \(H\) be a subgroup of \(F_n\). A function \(f \in F_n\) is said to be a piecewise-\(H\) function if there is a finite subdivision of the interval \([0, 1]\) such that on each interval \(J\) in the subdivision, \(f\) coincides with the restriction of some function \(h\in H\) to the interval \(J\). We note that since all breakpoints of elements in \(F_n\) are \(n\)-adic fractions, a function \(f\in F_n\) is a piecewise-\(H\) function if and only if there is an \(n\)-adic subdivision of the unit interval into finitely many \(n\)-adic sub-intervals such that on each \(n\)-adic interval in the subdivision, \(f\) coincides with some function in \(H\). Equivalently, a function \(f \in F_n\) is a piecewise-\(H\) function if and only if it has a (not necessarily reduced) tree-diagram \((T_1, T_2)\) such that each pair of branches \(u\to v\) of \((T_1, T_2)\) is a pair of branches of some element in \(H\). The following lemma was proved in [@G1] for the case \(n=2\). The proof for any \(n\) is similar.

The following remark follows directly from Lemma 19.

We will also make use of the following lemma, which was proved in [@G1] for \(F\) (see [@G1 Lemma 4.6]). The proof adapts readily to the general case; however, we provide an intuitive explanation below.

First, suppose there exists \(k \ge 0\) such that for every word \(w\) with \(|w| \ge k\), there is an element \(h \in H\) with the pair of branches \(uw \to vw\). By Lemma 19, it follows that there exists an element \(f \in \mathrm{Cl}(H)\) with the pair of branches \(u \to v\). So \(f\) is accepted by \(\mathcal{L}(H)\) and therefore \(u^+ = v^+\).

The other direction can be shown by induction on the iterative construction of the core \(\mathcal{L}(H)\). Recall that the core is obtained by identifying roots and leaves of tree-diagrams for a generating set, followed by a sequence of foldings. It can be shown that each type of folding preserves the validity of the claim: specifically, a type 1 folding leaves the value of \(k\) unchanged, while a type 2 folding increments it by 1.

2.3 On the derived subgroup \([F_n,F_n]\) {#sec:sec:preliminaries:F_n’}↩︎

Recall that \(F_n\) admits an infinite presentation [@B1] with generators \(x_0, x_1, x_2, \dots\) subject to the relations: \[x_k^{-1} x_i x_k = x_{i+n-1} \quad \text{for all } k < i.\]

The abelianization of \(F_n\), \(F_n / [F_n,F_n]\), is isomorphic to \(\mathbb{Z}^n\). Imposing commutativity on the generators yields the relations \(x_i = x_{i+n-1}\) for all \(i \ge 1\). Therefore, the image of any generator \(x_m\) (for \(m \ge 1\)) from the infinite generating set in the abelianization is simply \([x_j]\), where \(j \in \{1, \dots, n-1\}\) is the unique integer such that \(m \equiv j \pmod{n-1}\). Thus, the abelianization is the free abelian group generated by the \(n\) distinct equivalence classes \([x_0], [x_1], \dots, [x_{n-1}]\). The derived subgroup \([F_n,F_n]\) is precisely the kernel of the natural projection onto this free abelian group.

Recall that the elements of \(F_n\) act as piecewise-linear homeomorphisms of \([0,1]\). Since \(0\) and \(1\) are fixed points, the slope of any commutator must be trivial at \(0\) and \(1\). This motivates the following definition.

Note that for each \(n\), \([F_n,F_n]\leq \widetilde{F_n}\) (by the chain rule for derivatives at \(0^+\) and \(1^-\)), and that \(\widetilde{F}_2=[F_2,F_2]\). In fact, the following is the standard abelianization map of \(F_2\): \[f\rightarrow(\log_2f'(0^+),\log_2f'(1^-))\]

We will now show that \(\widetilde{F}_n\) is precisely the closure of the derived subgroup of \(F_n\). To do so, we use the following Lemma. Although it is well-known, we include a proof for completeness.

As a corollary of Lemma 23, we obtain the following well-known result.

3 On the generation problem {#sec:The generation problem}↩︎

3.1 Sufficient Conditions for Generation↩︎

In this section, we follow [@G1 section 7] closely, adapting most of the results for \(F_n\).

Let \(X\) be a finite subset of \(F_n\). We are interested in determining whether \(X\) generates \(F_n\). Let \(H:=\langle X\rangle\) be the subgroup of \(F_n\) generated by \(X\). We make the following observation: \(H=F_n\) if and only if both \(H[F_n,F_n]=F_n\) and \([F_n,F_n]\subseteq H\). Verifying the first condition is the same as checking if the image of \(X\) under the abelianization map generates \(\mathbb{Z}^n\) - and this is decidable. Thus, for the rest of this section, we focus on determining whether \([F_n,F_n]\subseteq H\). We start with a condition to check whether \([F_n,F_n]\subseteq\mathrm{Cl}(H)\).

Conditions (1) and (2) of Lemma 26 can be reformulated in a way that is easier to verify algorithmically. We call a vertex \(x\) in an \(n\)-ary tree-automaton an inner vertex if there is a rooted path \(p\) such that \(lab(p)\) is an inner \(n\)-ary word and \(p^+=x\). We then obtain the following corollary:

Note that by Corollary 27, the problem of determining whether the closure of the subgroup generated by a finite subset \(X \subseteq F_n\) contains \([F_n,F_n]\) is decidable.

The following lemma is well known (for example, a proof appears in [@Bleak2008]).

For an \(n\)-ary word \(s'\), we write \(.s'(n-1)^{\mathbb{N}}\) to denote the right end point of \([s']\). We prove the following two technical lemmas, which we will use later.

We will later use the following corollary from the last remark.

We provide another way of stating condition \((3)\) of the previous lemma.

We get the following theorem as a corollary of Lemma 39:

Note that as discussed earlier in this section, conditions \((1)\) and \((2)\) of Theorem 41 are verifiable. In the next section we will provide an algorithm that decides if condition \((3)\) holds. We do not have an algorithm deciding whether condition \((4)\) holds. However, we provide a stronger condition which is verifiable.

Recall that given a subset \(X\) of \(F_n\), the semi-core of \(X\), denoted by \(\mathcal{L}_{sem}(X)\), is an automaton constructed in a similar way to the core of \(X\). Namely, we identify all the roots and we identify all the pairs of leaves at the end of the pairs of branches of elements of \(X\). Finally, we apply all possible foldings of type 1.

Note that given a subset \(X\) of \(F_n\), \(\mathcal{L}_{sem}(X)\) might not be an \(n\)-ary tree-automaton (since foldings of type \(2\) may be applicable to \(\mathcal{L}_{sem}(X)\)). However, it is true that every \(n\)-ary word labels at most one (rooted) path on the semi-core of \(X\), so the notion of a vertex at the end of a path can be defined like in an \(n\)-ary tree-automaton (i.e., if \(u\) is an \(n\)-ary word which labels a path on \(\mathcal{L}_{sem}(X)\), \(u^+\) is the vertex at the end of \(u\)). The following claim was shown in [@G1] (it was shown for the case of \(F\), the proof for \(F_n\) is nearly identical).

In particular, by Lemma 26 and Remark 43 if \(X\) is a subset of \(F_n\), such that the closure of the group generated by \(X\) contains the derived subgroup of \(F_n\) then every \(n\)-ary word labels a path on \(\mathcal{L}_{sem}(X)\). We conclude the subsection by proving Theorem 1. Recall the Theorem, restated in an equivalent way.

Note that if \(X\) is finite, then \(\mathcal{L}_{sem}(X)\) is finite, and then condition \((4)\) of Theorem 44 is verifiable. In the next subsection we provide an algorithm for checking condition \((3)\) of Theorem 44.

3.2 The Tuples Algorithm↩︎

In this section, we present an algorithm verifying condition \((3)\) of Theorem 41. We outline the definitions and claims needed to build the algorithm. We omit the supporting claims and proofs, which are very similar to those found in Section 8 of [@G1].

For each \(i\in\{0,\dots,n-2\}\), we define the set: \[S_H^i := \{ (a, b) \mid \exists h \in H,\ \exists\alpha \in D_{n,i} \text{ such that } \log_n{h'(\alpha^-)} = a,\ \log_n{h'(\alpha^+)} = b,\ \text{and } h(\alpha) = \alpha \}\] One can show that \(S_H^i\) is a subgroup of \(\mathbb{Z}\times\mathbb{Z}\), and that \((0,1)\in S_H^i\) if and only if \((1,0)\in S_H^i\). Thus, Problem 45 is equivalent to verifying whether \(S_H^i=\mathbb{Z}\times\mathbb{Z}\) for all \(i\in\{0,\dots,n-2\}\).

\(\mathcal{T}_H\) naturally forms a groupoid under a partial addition operation and an inverse operation. Namely, given two tuples \(t_1=(a,b,[u]_X\rightarrow[v]_X,i\rightarrow j)\) and \(t_2=(c,d,[v]_X\rightarrow[w]_X,j\rightarrow k)\), we define \(t_1+t_2=(a+c,b+d,[u]_X\rightarrow[w]_X,i\rightarrow k)\) and \(t_1^{-1}=(-a,-b,[v]_X\rightarrow[u]_X,j\rightarrow i)\).

Tuples of the form \((a,b,[u]_X\rightarrow[u]_X,i\rightarrow i)\) are called spherical tuples. For a fixed word \(u\) and index \(i\), the set of spherical tuples forms a group, denoted \(\mathcal{T}_H([u]_X,i)\). We define a group homomorphism \(\Psi: \mathcal{T}_H([u]_X,i) \rightarrow \mathbb{Z}\times\mathbb{Z}\) by \(\Psi(a,b,[u]_X\rightarrow[u]_X,i\rightarrow i)=(a,b)\).

The following core propositions allow us to verify Problem 45 algorithmically.

Using Proposition 49, we formulate a procedure to decide if \(S_H^i = \mathbb{Z}\times\mathbb{Z}\) for all \(i\), completing the verification of condition (3) of Theorem 41.

The Algorithm↩︎

Input: A finite generating set \(X\) for \(H\leq F_n\) satisfying \([F_n,F_n]\subseteq\mathrm{Cl}(H)\).

Output: True if \(S_H^i=\mathbb{Z}\times\mathbb{Z}\) for all \(i\in\{0,\dots,n-2\}\); otherwise False.

1. Compute the finite automaton \(\mathcal{L}_{sem}(X)\) and let \(N\) be its number of vertices.

2. Initialize \(Y\) as the set of all tuples associated with the reduced tree-diagrams of elements of \(X^{\pm 1}\).

3. Add all trivial spherical tuples \((0,0,[u]_X\rightarrow [u]_X,i\rightarrow i)\) to \(Y\), for every equivalence class \([u]_X\) represented by a vertex of \(\mathcal{L}_{sem}(X)\) and every \(i\in\{0,\dots,n-2\}\).

4. Compute \(M'(i)\), the set of all spherical tuples in \(\mathcal{T}_H([\emptyset]_X,i)\) generated by composing at most \(N\) elements from \(Y\).

5. Compute \(M(i)=\{\Psi(t)\mid t\in M'(i)\}\).

6. If, for each \(i\in\{0,\dots,n-2\}\), the set \(M(i)\) generates \(\mathbb{Z}\times\mathbb{Z}\), return True; otherwise return False.

4 On core automata↩︎

Our goal in this section is to develop several tools regarding automata, which we will use to prove Theorem 2. We start by introducing a slight generalization of the rooted \(n\)-ary tree automata from Subsection 2.2. Specifically, we define a rooted tree automaton similarly to a rooted \(n\)-ary tree automaton, but without requiring that each father vertex has the same number of children.

In our setting, a tree is a connected, acyclic graph. A rooted tree \(\mathcal{T}\) is a directed tree with a unique distinguished vertex \(r\), called the root, such that there exists a unique directed path from \(r\) to every other vertex in \(\mathcal{T}\) (equivalently, \(r\) has in-degree \(0\) and all other vertices have in-degree \(1\)). For any vertex \(v \in \mathcal{T}\) with out-degree \(n(v) > 0\), its outgoing edges are naturally labeled by \(0, \dots, n(v)-1\).

Given a rooted tree \(\mathcal{T}\), a subgraph \(\mathcal{T}'\) is called a full sub-tree of \(\mathcal{T}\) if it is a rooted sub-tree sharing the same root \(r\), and it satisfies the following condition: for any vertex \(v \in \mathcal{T}'\) that is not a leaf in \(\mathcal{T}'\) (i.e., \(v\) has at least one outgoing edge in \(\mathcal{T}'\)), all of its children and outgoing edges in \(\mathcal{T}\) are also contained in \(\mathcal{T}'\).

Let \(\mathcal{A}_r\) be a rooted semi tree-automaton. We follow [@G1; @G2] to construct rooted trees \(\mathcal{T}_{\mathcal{A}_r},\mathcal{T}_{\mathcal{A}_r}^{min}\) associated with \(\mathcal{A}_r\).

:::: {#fig:tree-automaton example .figure} ::: caption An example of a rooted tree-automaton \(\mathcal{A}_r\), the path tree \(\mathcal{T}_{\mathcal{A}_r}\) and a minimal rooted tree \(\mathcal{T}_{\mathcal{A}_r}^{min}\) associated with \(\mathcal{A}_r\). Each vertex in those trees, corresponding to a path \(p\) in the automaton, is labeled by the end vertex of \(p\). Every caret in each of the trees, rooted in some vertex \(p\), corresponds to the outgoing edges of the vertex \(p^+\) in the automaton. ::: ::::

Each rooted tree-automaton gives rise to a semigroup presentation. We call such a semigroup presentation a tree semigroup presentation. Alternatively, such presentations can be defined axiomatically:

These constructions are inverses of each other. More precisely, if \(\mathcal{A}_r\) is a rooted semi tree-automaton then \(\mathcal{A}(\mathcal{P}(\mathcal{A}_r))=\mathcal{A}_r\), and if \(\mathcal{P}_r\) is a semi tree semigroup presentation then \(\mathcal{P}(\mathcal{A}(\mathcal{P}_r))=\mathcal{P}_r\).

Notice that if \(\mathcal{P}_r\) is a tree semigroup presentation, then \(\mathcal{A}(\mathcal{P}_r)\) is a rooted tree-automaton. Also, if \(\mathcal{A}_r\) is a rooted tree-automaton, then \(\mathcal{P}(\mathcal{A}_r)\) is a tree semigroup presentation.

Note that if \(\mathcal{Q}_r\) is a basic splitting of \(\mathcal{P}_r\) then, in the notation of Definition 56 there is a natural bijection between paths on \(\mathcal{A}(\mathcal{Q}_r)\) that do not end in the vertex \(x\) and paths on \(\mathcal{A}(\mathcal{P}_r\)). A way to see this is by considering a path as the sequence of vertices it visits. Given a path \(q\) in \(\mathcal{A}(\mathcal{Q}_r)\), its image under the bijection is the path in \(\mathcal{A}(\mathcal{P}_r)\) obtained by removing all occurrences of the vertex \(x\) from \(q\).

The validity of the following claim is evident.

The following claim can be easily verified, and we use it to prove the lemma that follows.

We state two definitions and a lemma that were given in [@G2] for \(F_2\). The proof of the lemma readily adapts to the case of \(F_n\).

As a corollary of Lemmas 58, 60 and 63 we get the following.

Let \(\mathcal{A}_r\) be a rooted \(n\)-ary semi tree automaton (i.e., a rooted semi tree automaton where each father vertex has exactly \(n\) children). Let \(f\in F_n\). We say that \(f\) is accepted by \(\mathcal{A}_r\) if there exists a tree diagram representing \(f\), which is accepted by \(\mathcal{A}_r\) (where acceptance of a tree-diagram by a rooted \(n\)-ary semi tree automaton is defined to be the same as acceptance of a tree-diagram by a rooted \(n\)-ary tree automaton). The diagram group accepted by \(\mathcal{A}_r\) is defined as the group of all elements in \(F_n\) accepted by \(\mathcal{A}_r\), and is denoted by \(\mathcal{DG}(\mathcal{A}_r)\). Note that if \(\mathcal{A}_r\) is a rooted \(n\)-ary tree automaton, these definitions coincide with Definition 15.

5 Maximal subgroups↩︎

In this section, we answer a variant of a question posed by Savchuk [@Sav1; @Sav2], which was also asked by Aiello and Nagnibeda [@AN], regarding the existence of maximal subgroups of \(F_n\).

In [@Sav1], Savchuk initiated the study of maximal subgroups of \(F_2\) by proving that for any \(\alpha\in(0,1)\), the stabilizer of \(\alpha\) in \(F_2\) is a maximal subgroup of infinite index. He asked whether there are other maximal subgroups of infinite index in \(F_2\). Golan and Sapir answered this question affirmatively in [@GS1] by providing an explicit example of a maximal subgroup of infinite index in \(F_2\) that does not fix any point. They also introduced a general method for constructing maximal subgroups of \(F_2\) via core automata. Aiello and Nagnibeda [@AN2] found three additional examples of maximal subgroups of infinite index in \(F_2\) that do not fix any point in \((0,1)\). Later, in [@G2], Golan found a countable family of non-isomorphic maximal subgroups of infinite index in \(F_2\) that do not fix any point in \((0,1)\). Aiello and Nagnibeda extended Savchuk’s question to \(F_n\) for a general \(n\ge 2\), and provided an example of a maximal subgroup of infinite index in \(F_3\) that does not fix any point in the open unit interval ([@AN]). The question for a general \(n\) remained open.

We show that for every \(n\ge 2\), \(F_n\) has a maximal subgroup of infinite index that does not fix any point in \((0,1)\). Following the approach introduced by Golan and Sapir, we analyze subgroups accepted by core automata. We first note the following result, which is an immediate consequence of Theorem 44.

:::: {#fig:core of \(F_n\) .figure} ::: caption An image describing a minimal tree associated with the core of \(F_n\). The children of a vertex labeled by \(a_i\) are written separately of the tree. To see a minimal tree, one should think about those carets as a part of the tree. ::: ::::

We now prove Theorem 2. Recall the theorem.

::::::::

We show condition \((1)\) holds. Any element of \(F_n\) which has the pair of branches \(0\rightarrow 00\) (such an element clearly exists) is not accepted by \(\mathcal{A}_r\). Therefore, \(H\neq F_n\), so \(H\) is a strict subgroup of \(F_n\).

Note that since \(\mathcal{A}_r\) is a core automaton, it follows that \(\mathcal{L}(H)\) is equal to \(\mathcal{A}_r\). So \(H\) equals to the subgroup of \(F_n\) accepted by \(\mathcal{L}(H)\), which is \(\mathrm{Cl}(H)\).

We now show that condition \((a)\) holds. We need to show that \(\pi_{ab}(H)=\mathbb{Z}^n\). It is enough to find elements \(h_0,\dots,h_{n-1}\) of \(F_n\) that are accepted by \(\mathcal{A}_r\), such that their images under the abelianization map generate \(\mathbb{Z}^n\). Let \(x_0,\dots,x_{n-1}\) be the standard generators of \(F_n\). We define the following elements.

• \(h_0=x_0x_{n-1}x_0^{-1}\).

• for each \(1\le i\le n-2\), \(h_i=x_0x_ix_{n-1}x_{2n-2}^{-1}x_{n-1}^{-1}x_0^{-1}\).

• \(h_{n-1}=x_0^2x_{n-1}x_{2n-2}^{-1}x_{n-1}^{-1}x_0^{-1}\).

Note that they are accepted by \(\mathcal{A}_r\). See Figure [-@{fig:elements for abelianization}] for tree-diagrams of these elements. Consider the following presentation of \(F_n\): \[\langle x_0,x_1,\dots|x_j^{x_i}=x_{j+n-1} \text{ for }i<j\rangle\] For natural \(i\ge 0\), denote \([x_i]\) the coset of \(x_i\) in the abelianization. Recall that for \(i\ge1\), \([x_i]=[x_{i+n-1}]\), so the abelianization of \(F_n\) is generated by \([x_0],\dots,[x_{n-1}]\). Note that

• \([h_0]=[x_{n-1}]\).

• for each \(1\le i\le n-2\), \([h_i]=[x_i][x_{n-1}^{-1}]\).

• \([h_{n-1}]=[x_0][x_{n-1}^{-1}]\).

So \([h_0],\dots,[h_{n-1}]\) generate the abelianization, and condition \((a)\) holds.

:::::: {#fig:elements for abelianization .figure} ::: minipage :::

::::::

We move onto condition \((b)\). We take the elements \(h_0,\dots,h_{n-2}\in H\). Let \(\alpha_0=.0(n-1)=\frac{n-1}{n^2}\), and for \(1\le i\le n-2\), let \(\alpha_i=.0i=\frac{i}{n^2}\). Note that for each \(i\in\{0,\dots,n-2\}\), we have \(\alpha_i\in D_{n,i}\), and \(h_i(\alpha_i)=\alpha_i,h_i'(\alpha_i^-)=1,h_i'(\alpha_i^+)=n\), so condition \((b)\) holds.

We move to conditions \((c)\) and \((d)\). Let \(g_1\in F_n\setminus H\). Recall that \(H=\mathrm{Cl}(H)\) and that every \(n\)-ary word labels a path on \(\mathcal{L}(H)\). So there are \(n\)-ary words such that \(g_1\) has a pair of branches \(u\rightarrow v\) such that \(u^+\neq v^+\) on \(\mathcal{L}(H)\). Therefore, in \(\mathcal{L}_{sem}(H\cup\{g_1\})\), we identify the vertices \(u^+\) and \(v^+\). Note that in particular, \(t(u)=t(v)\). Also note that since for any \(k\in\mathbb{N}\setminus\{0\}\), \(((n-1)^k)^+=g\), neither \(u\) nor \(v\) can be of the form \((n-1)^k\) (since if one of them is of this form, the other must be of the form \((n-1)^\ell\) and then they both end in the vertex \(g\) in \(\mathcal{A}_r\)). Also, if \(u\) or \(v\) are of the form \(0^k\) for some \(k\in\mathbb{N}\setminus\{0\}\), then we must have \(\{u^+,v^+\}=\{f,f_0\}\). So \(u^+\neq v^+\), and one of the following must hold:

• \(u^+,v^+\in\{f_0,f\}\)

• \(u^+,v^+\in\{a_0,a_{n-1},c_0,c_{n-1}\}\)

• \(u^+,v^+\in\{a_i,a_{i+n-1}\}\) for some \(1\le i \le n-2\)

Consider \(\mathcal{L}_{sem}(H\cup\{g_1\})\). One can construct it by taking \(\mathcal{L}(H)\), identifying \(u^+\) and \(v^+\), and the other end vertices of the pairs of branches of \(g_1\), and apply foldings of type \(1\). We will show that in each case between \((1),(2),(3)\), there is an \(n\)-ary word \(w\) such that for every two \(n\)-ary words \(w_1,w_2\) such that \(t(w_1)=t(w_2)\), the words \(ww_1,ww_2\) have \((ww_1)^+=(ww_2)^+\) on \(\mathcal{L}_{sem}(H\cup\{g_1\})\), showing condition \((c)\) holds. We will also show that after applying foldings of type \(2\) as well, we get the core of \(F_n\), proving condition \((d)\) holds.

We now construct a directed graph \(G(\mathcal{A}_r)=(V,E)\) that simulates the identifications of vertices in the automaton after foldings of type \(1\). The set of vertices is the set of all possible pairs of vertices we identify. Namely, \[\begin{align} V=\{(f_0,f),(c_0,c_{n-1}),(c_0,a_0),(c_0,a_{n-1}),(c_{n-1},a_0),\\(c_{n-1},a_{n-1}),(a_0,a_{n-1}),(a_1,a_n),\dots,(a_{n-2},a_{2n-3})\} \end{align}\]We add an edge from the vertex \((u_1,v_1)\) to the vertex \((u_2,v_2)\) if the identification of \((u_2,v_2)\) follows from the identification of \((u_1,v_1)\) by applying a single folding of type \(1\). Let us first consider the case where \(n\ge3\). See Figure 7 for the graph.

Note that there is a path between any vertex in \(V\) to any vertex in
\(V\setminus\{(c_{n-1},a_{n-1}),(c_{n-1},a_0),(f,f_0)\}\). So in any case between \((1),(2),(3)\), the identification \(u^+=v^+\) will result in the identifications of \(c_0=c_{n-1}=a_0=a_{n-1}\) and \(a_i=a_{i+n-1}\) for each \(i\in\{1,\dots,n-2\}\) after some foldings of type \(1\). Therefore \(\mathcal{L}_{sem}(H\cup\{g_1\})\) has exactly \(n-1\) inner vertices. Since every \(n\)-ary word labels a path on \(\mathcal{L}(H)\), every \(n\)-ary word also labels a path on \(\mathcal{L}_{sem}(H\cup\{g_1\})\). In particular, for the word \(w=1\), for every \(n\)-ary words \(w_1,w_2\) such that \(t(w_1)=t(w_2)\) it holds that \(ww_1,ww_2\) label paths on \(\mathcal{L}_{sem}(H\cup\{g_1\})\) and \((ww_1)^+=(ww_2)^+\) on \(\mathcal{L}_{sem}(H\cup\{g_1\})\). So condition \((c)\) is satisfied.

Also note that the children of \(f\) and \(f_0\) will be the same in \(\mathcal{L}_{sem}(H\cup\{g_1\})\), so \(f\) and \(f_0\) will be identified in \(\mathcal{L}(H\cup\{g_1\})\) by a folding of type \(2\). So \(\mathcal{L}(H\cup\{g_1\})=\mathcal{L}(F_n)\), so it accepts the derived subgroup of \(F_n\), and condition \((d)\) holds.

Finally, if \(n=2\), we get that \(a_0,a_{n-1}\) and \(c_0\) must be identified after applying foldings of type \(1\). The children of \(a_0\) and \(c_{n-1}\) will be the same, so after applying foldings of type \(2\) as well, the new automaton will accept \([F_n,F_n]\), so condition \((d)\) holds. By results of Golan ([@G1]) for the case of \(F_2\), we do not have to show condition \((c)\), and we will get that \(\langle H,g\rangle=F_2\), so we are done.

This shows that condition \((1)\) holds.

Finally, for condition \((2)\), we need to show that \(H\) is of infinite index, and it does not fix any point in the open unit interval. To show the first claim, note that since \([F_n,F_n]\) is a simple subgroup of infinite index of \(F_n\) ([@Higman1974; @B1]), it follows that the subgroups of finite index of \(F_n\) contain the commutator \([F_n,F_n]\). So, if \(H\) was a subgroup of finite index of \(F_n\), then by Corollary 27, \(\mathcal{L}(H)=\mathcal{A}_r\) would have exactly \(n-1\) inner vertices. Since \(\mathcal{A}_r\) has \(2n\) inner vertices, \(H\) is not a subgroup of finite index of \(F_n\).

For the latter claim, note that for any inner \(n\)-ary word \(u\), there exists some inner \(n\)-ary word \(v\) such that \(u^+=v^+\) and the intervals \([u]\) and \([v]\) are disjoint. To see this, note that for every inner vertex \(a\) of \(\mathcal{A}_r\), there exists a path ending in \(a\) in \(\mathcal{A}_r\) whose label starts with the string \(00\), and there exists another path ending in \(a\) in \(\mathcal{A}_r\) whose label starts with \(n-1\). So, for every point \(x\in (0,1)\), we can find some inner \(n\)-ary word \(u\) such that \(x\in [u]\), and an element \(h\in H\) such that \(h\) maps \([u]\) to a disjoint interval, and in particular \(h\) does not fix \(x\). ◻ :::

6 Further research and subsequent developments↩︎

We conclude with several questions suggested by the generation criterion proved in this paper, and with some comments on subsequent developments.

Theorem 44 gives a sufficient criterion for a finite set \(X\subseteq F_n\) to generate \(F_n\). The criterion consists of the closure condition, the abelianization condition, the tuple condition, and the semi-core condition. The first open problem is whether the semi-core condition is superfluous.

A positive answer would give a solution to the generation problem in \(F_n\). Indeed, in that case, conditions \((1)\), \((2)\) and \((3)\) would be necessary and sufficient for \(H\) to be equal to \(F_n\), and they are effectively checkable for finitely generated subgroups using the core, the abelianization, and the tuple algorithm.

For Thompson’s group \(F=F_2\), the answer to Problem 69 is positive [@G1]. This yields an effective necessary and sufficient criterion for generation in \(F\). In subsequent work on maximal subgroups of \(F\) [@G2], it was proved that, for subgroups of \(F=F_2\), the conditions \((1)\) and \((2)\) already imply condition \((3)\). Thus, in \(F\), there is an even shorter criterion: the closure and the image in the abelianization determine whether a subgroup is the whole group.

It is natural to ask whether the same short criterion holds for the groups \(F_n\).

A positive answer to Problem 70 would remove the tuple condition from the final statement of the generation criterion and would give a short generation test for \(F_n\), analogous to the criterion known for \(F\).

Problem 70 is also closely related to maximal subgroups of infinite index. In fact, a positive answer is equivalent to the assertion that every maximal subgroup of infinite index of \(F_n\) is closed.

Indeed, assume first that Problem 70 has a positive answer, and let \(M\) be a maximal subgroup of infinite index in \(F_n\). In particular, \(M\) is not contained in any proper finite index subgroup of \(F_n\). Hence, \(M[F_n,F_n]=F_n\). If \(M\) is not closed, then by maximality \(\mathrm{Cl}(M)=F_n\). Hence \(M\) satisfies the two hypotheses of Problem 70, and so \(M=F_n\), a contradiction. Therefore \(M\) is closed.

Conversely, suppose that every maximal subgroup of infinite index in \(F_n\) is closed. Let \(H< F_n\) be a strict subgroup such that \[[F_n,F_n]\subseteq \mathrm{Cl}(H) \quad\text{and}\quad H[F_n,F_n]=F_n.\] Then \(\mathrm{Cl}(H)=F_n\). Moreover, since \(H[F_n,F_n]=F_n\), \(H\) is not contained in any proper finite-index subgroup of \(F_n\). Since \(F_n\) is finitely generated, Zorn’s lemma implies that \(H\) is contained in a maximal subgroup \(M\) of \(F_n\). This maximal subgroup cannot have finite index, and hence is closed by assumption. But then \[F_n=\mathrm{Cl}(H)\leq \mathrm{Cl}(M)=M,\] a contradiction. Thus no such proper subgroup \(H\) exists, and Problem 70 has a positive answer.

Therefore the short generation criterion in Problem 70 holds if and only if every maximal subgroup of infinite index of \(F_n\) is closed. This equivalence is one of the reasons the problem is important: it connects the generation problem directly to the structure of maximal subgroups.

The question whether every maximal subgroup of \(F_n\) of infinite index is closed is relevant to a broader classification problem for maximal subgroups of Higman–Thompson groups. After the completion of the present work, the first author formulated the following problem.

The point-stabilizer examples are excluded by the minimality hypothesis. Problem 71 is consistent with all examples of maximal subgroups of \(F\) which appear in the literature. More generally, the same pattern occurs in the known examples of maximal subgroups of infinite index in Higman–Thompson groups which do not arise as point stabilizers. The maximal subgroups constructed in the present paper are isomorphic to \(F_{2n-1}\). In addition, after the completion of the present work, the first author proved that the Jones subgroup of \(F_3\) used by Aiello and Nagnibeda is isomorphic to \(F_4\) (the result will appear in a future paper).

If every maximal subgroup of infinite index of \(F_n\) is closed, then the study of closed subgroups becomes even more central to the classification of maximal subgroups. This motivates the following broader problems.

Several results obtained by the first author after the completion of this paper support this direction. The first author has proved that a wide class of closed subgroups of \(F\), and more generally of \(F_n\), are isomorphic to Higman–Thompson groups. These results apply, in particular, to the diagram groups associated with the classes \(\mathcal{C}_n\) of indecomposable geometrically fast groups generated by \(n\) one-bump functions. For the terminology of geometrically fast sets, dynamical diagrams, and the classes \(\mathcal{C}_n\), see [@BBKMZ; @BelkStott]. Belk and Stott proved that fast one-bump groups are isomorphic to diagram groups determined by their dynamical diagrams, and used this description to prove that pseudo-\(F_4\) is isomorphic to \(F_4\) [@BelkStott]. Although the original fast one-bump groups are not, in general, closed subgroups of \(F\), the corresponding diagram groups can be realized as closed subgroups of \(F\) using methods from [@GSclosed]. The first author proved that, for every \(n\geq 2\), every group in the class \(\mathcal{C}_n\) is isomorphic to \(F_n\). The cases \(n=2,3\) were previously known, and the case \(n=4\) is the theorem of Belk and Stott. Thus, for the classes \(\mathcal{C}_n\), this answers the strong form of a question of Brin and Zaremsky from [@BurilloBuxNucinkis2018]. The first author has also constructed, for every pair \(m,n\) with \(n\geq m\geq 2\), a maximal subgroup of \(F_m\) isomorphic to \(F_n\). The proofs of these results will appear in subsequent papers.

These later results make essential use of the generation criterion and core methods developed in the present paper, together with additional arguments.

Gili Golan
Department of Mathematics,
Ben-Gurion University of the Negev,
golangi@bgu.ac.il

Eytan Sapir
Department of Mathematics,
Ben-Gurion University of the Negev,
esapir@post.bgu.ac.il

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1. This subgroup is denoted \(\vec{F}_3\) in [@AN]. This notation should not be confused with the subgroup \(\vec{F}_3\) of \(F\) appearing in the family \(\vec{F}_n\) in [@G2].↩︎