2·1. Ważewski dendrites

We begin with the general definition of dendrites, then we introduce a concept of order of a point which will be useful to define what Ważewski dendrites are.

There are many equivalent definitions of dendrites. For example, we could also define a dendrite to be a non-empty compact connected metric space such that the intersection of any two connected subsets is a connected subset ([ Reference S. B.37 , theorem 10·10]). See also [ 50 , section V.1, (1·1) and (1·2)] for further characterisations of dendrites among continua and locally connected continua. Non-degenerate dendrites are also the underlying topological spaces behind the so-called metric trees ([ 11 , Proposition 2·2]). Examples of dendrites are the compactification of any simplicial tree (as noted in [ 22 , Proposition 12·2]) and certain Julia sets (for instance, for the complex map $z \to z^2 + i$ ).

The order of a point p of a dendrite X is the cardinality of the set of connected components of $X \setminus \{p\}$ . (For dendrites, this is equivalent to the more general notion of Menger–Urysohn order, which is discussed in [ Reference Kuratowski33 , 7, section 51, pp. 274–307]; see also [ Reference S. B.37 , Definition 9·3].) A distinction of points of a dendrite based on their order will be given later in Subsection 3·2.

For the purpose of this work, we are going to consider the most regular non-trivial dendrites, which are called Ważewski dendrites and are defined right below. We would like to mention that a general dendrite might be so irregular that its homeomorphism group is trivial, so it is natural that the most regular dendrites would produce larger and often more interesting groups of homeomorphisms.

For example, Figure 1 depicts the Ważewski dendrite of order 3.

By [ 18 , theorem 6·2], $D_n \simeq D_m$ if and only if $n = m$ , so we can actually talk about the Ważewski dendrite of order n. In the next Subsections we will build these dendrites as limit spaces of replacement systems.

Fig. 2. The Airplane replacement system.

2·2. Replacement systems

We will be working with finite graphs, all of which are directed edge-coloured graphs where we allow both loops and parallel edges. Thus, for our purpose a graph will be a sixtuplet $(V, E, C, \mathrm{i}, \mathrm{t}, \mathrm{c})$ where:

1. (i) V, E and C are finite sets, whose elements are called vertices, edges and colours, respectively;

2. (ii) i and t are maps $E \to V$ , whose images $\mathrm{i}(e)$ and $\mathrm{t}(e)$ are called the initial vertex and the terminal vertex of the edge e;

3. (iii) c is a map $E \to C$ , and each image $\mathrm{c}(e)$ is called the colour of the edge e.

When the set of colours is a singleton (which will be the case for dendrite replacement systems until Subsection 8·2), the map c is trivial, and both the set C and the map c can be naturally omitted. In this case, we say that the replacement system is monochromatic.

We say that an edge e is incident on a vertex v when $\iota(e)=v$ or $\tau(e)=v$ . When two edges are incident on a common vertex, we say that they are adjacent.

Starting from the base graph $\Gamma$ , we can expand it by replacing one of its edges e with the c(e) replacement graph, attaching the initial and terminal vertices of $R_{c(e)}$ in place of the initial and terminal vertices i(e) and t(e) of the edge e. The same procedure can be applied to any expansion of the base graph, which allows to obtain infinitely many further expansions of the base graph. For example, Figure 2 shows the Airplane replacement system (which realises the Airplane rearrangement group $T_A$ from [ Reference Belk and Forrest5 , example 2·13], studied in [ Reference Tarocchi49 ]), and Figure 3 portrays two subsequent expansions of the base graph of this replacement system: the first one is a blue expansion (and it is the only possible one, because the base graph consists of a sole edge) and the second one is a red expansion. The order of expansions does not matter when expanding two edges of the same graph (for example, if we want to expand the edges $ib_1$ and $ib_3$ in one of the two graphs depicted in Figure 3, it does not matter which of the two we expand first.

Fig. 3. Two subsequent expansions of the base graph of the Airplane replacement system.

If we name each edge of the base and replacement graphs by distinct symbols in an alphabet $\mathbb{A}$ , then we can represent the edges of an expansion of the base graph as certain finite words in the alphabet $\mathbb{A}$ . Figure 3 shows, for example, that the edges newly obtained by performing the expansions are words of length two or three (in this case the first letter is always i because i is the only edge of the base graph, so it’s the only possible first expansion).

2·3. Limit spaces

The limit space of a replacement system is built as a quotient of a Cantor space $\Omega$ by an equivalence relation. Here are their definitions.

In essence, an element of $\Omega$ is an infinite sequence of edges, each “nested” below the previous one via expansions. This space is actually a one-sided shift of finite type, but there is no need to be familiar with that machinery to have a clear understanding of what $\Omega$ is.

We will only be considering replacement systems that satisfy the natural requirements expressed in the next definition.

When equipped with the subspace topology inherited from the product topology of $\mathbb{A}^\infty$ (which is a Cantor space), the symbol space $\Omega$ is always compact, metrisable and totally disconnected. Indeed, it inherits a basis of clopen sets from the cones of $\mathbb{A}^\infty$ (a cone is the subspace consisting of all of those sequences that have a fixed common prefix). Then $\Omega$ is compact because it is an intersection of cones and it is metrisable and totally disconnected because $\mathbb{A}^\infty$ is. Additionally, when the replacement system is expanding, $\Omega$ is a Cantor space, since it has no isolated point because every cone contains multiple elements of $\Omega$ by Definition 2·5(3).

Definition 2·6. Given a replacement system $\mathcal{R}$ and its symbol space $\Omega$ , we define the gluing relation $\sim$ as

\[ \alpha \sim \beta \iff \alpha_1 \dots \alpha_n \text{ and } \beta_1 \dots \beta_n \text{ are equal or adjacent edges in } \Gamma_n, \forall n \in \mathbb{N}, \]

where $\alpha = \alpha_1 \alpha_2 \dots$ and $\beta = \beta_1 \beta_2 \dots$ are sequences in $\Omega$ , and $\Gamma_n$ is the sequence of graphs obtained by expanding, at each step, every edge of the previous graph, starting from the base graph $\Gamma$ (thus, $\Gamma_n$ contains precisely those edges that correspond to allowed finite words of length n).

By [ Reference Belk and Forrest5 , proposition 1·9], the gluing relation defined by an expanding replacement system is an equivalence relation, so the following definition makes sense.

Continuing with our example, the limit space of the Airplane replacement system, simply called the Airplane limit space, is depicted in Figure 4. It is homeomorphic to the Julia set for the complex map $z \mapsto z^2 - 1.755$ . In Subsection 3·1 we will build replacement systems whose limit spaces are Ważewski dendrites.

Fig. 4. The Airplane limit space.

By [ Reference Belk and Forrest5 , Theorem 1·25], limit spaces are compact metric spaces. We remark here that, while fractals are usually thought of subsets of a Euclidean space, the limit spaces produced here are only defined as topological spaces. For example, under this point of view the famous Koch snowflake is to be thought of as topologically equivalent to a circle.

2·4. Rearrangement groups

Given an expanding replacement system $\mathcal{R}$ and its limit space X, consider an edge e of some expansion of the base graph (i.e., any allowed finite word). We define the cell $\boldsymbol{C(e)}$ as the image of the cone of those sequences of the symbol space that have e as a prefix via the quotient map of the gluing relation. It will often be useful to consider the topological interior of a cell C(e), which will be denoted by $C^{\mathrm{o}}(e)$ .

Each cell C(e) has a type, distinguished by whether the edge e is a loop or not and on its colour. For example, the Airplane limit space admits two types or cells, only distinguished by their colour, as there are no loops among the expansions of the base graph. Two cells $C(e_1)$ and $C(e_2)$ of the same type admit a natural canonical homeomorphism, which is given by the map that changes the prefix $e_1$ with $e_2$ . Depending on whether it is a loop or not, a cell C(e) has one or two endpoints, which correspond to the initial and terminal vertices of the edge e.

To each expansion $\Lambda$ of the base graph we associate a cellular partition of the limit space X (i.e., a cover of X by finitely many cells whose interiors are disjoint, meaning that they may only intersect at the endpoints of the cells) as follows:

\[ \Lambda \mapsto \{ C(e) \mid e \text{ is an edge of } \Lambda \}, \]

which is a bijection between the set of expansions of the base graph and that of cellular partitions. The notions of canonical homeomorphisms and cellular partitions prompt the following definition:

It is not hard to see that rearrangements form a group under composition, so we can talk about rearrangement groups of a given expanding replacement system.

Rearrangements naturally correspond to graph isomorphisms (which need to preserve edge orientation and colours) between two expansions of the base graph: if the graph isomorphism maps an edge $e_1$ to $e_2$ , then the rearrangement restricts to a canonical homeomorphism from the cell $C(e_1)$ to $C(e_2)$ . A triple (D, R, f) given by two expansions D and R of the base graph and a graph isomorphism $f\,:\, D \to R$ is called a graph pair diagram for the induced rearrangement. Each rearrangement has multiple graph pair diagrams that represent it (one can always “expand” a graph pair diagram by expanding an edge e in the domain graph and its image under the graph isomorphism in the range graph), but there always exists a unique reduced graph pair diagram for each rearrangement.

Finally, we remark here that rearrangement groups act properly by isometries on CAT(0) cubical complexes, as explained in [ Reference Belk and Forrest5 , section 3]. In particular, an application of [ Reference Chatterji and Drutu19 , Theorem 1·2] to the action of a rearrangement group on the 1-skeleton of this complex yields that rearrangement groups do not have Kazhdan’s Property (T), nor do any of their infinite subgroups (such as the commutator subgroup of dendrite rearrangement groups).

2·5. Thompson groups

Thompson’s group F is the group of those orientation-preserving homeomorphisms of the unit interval [0, 1] that are piecewise linear with slopes that are powers of 2 and breakpoints that are dyadic (i.e., they can be written as fractions whose denominator is a power of 2, or equivalently they belong to $\mathbb{Z}[{1}/{2}]$ ). Thompson groups T and V can be defined in a similar fashion, with the difference that T acts on the unit circle and V on the Cantor space in place of the unit interval. We refer to [ Reference Cannon, Floyd and Parry15 ] for more general information about Thompson groups, while [ Reference Belk2 ] is specifically about F.

As mentioned earlier, Thompson groups F, T and V can all be realised as rearrangement groups; Figure 5 depicts a replacement system for F, which in the following is going to be featured much more prominently than the other two. In particular, we are going to frequently use the transitivity properties of F on the set of dyadic points of [0, 1] and the fact that F is generated by the two elements portrayed in Figure 6. Moreover, in Section 6 we are going to use the fact that the commutator subgroup [F, F] consists precisely of those elements of F that act trivially in some neighbourhood of 0 and of 1.

Fig. 5. A replacement system for Thompson’s group F.

Fig. 6. The generators $X_0$ and $X_1$ of Thompson’s group F.

3·1 Replacement systems for dendrites

In this Subsection we define the replacement systems $\mathcal{D}_n$ that produce the Ważewski dendrites $D_n$ as limit spaces, for any natural number $n \geq 3$ . Figure 7 schematically depicts the generic replacement system $\mathcal{D}_n$ , which is defined with more precision as follows. The dendrite replacement system $\boldsymbol{\mathcal{D}_n}$ is the monochromatic replacement system whose base and replacement graphs are the same tree consisting of a vertex of degree n that is the origin of n edges terminating at n distinguished leaves; the initial and terminal vertices $\iota$ and $\tau$ of the replacement graph are two distinct leaves.

Fig. 7. A schematic depiction of the dendrite replacement system $\mathcal{D}_n$ (the dotted lines are meant to indicate that each graph has a total of n edges).

The replacement system $\mathcal{D}_n$ is expanding (Definition 2·5), so the gluing relation is an equivalence relation [ Reference Belk and Forrest5 , propositions 1·9 and 1·22] and the limit space of $\mathcal{D}_n$ is defined as in Definition 2·7. When endowed with the quotient topology induced by the gluing relation (see [ Reference Belk and Forrest5 , subsection 1·5]), the limit space is a non-empty compact metrisable space (by [ Reference Belk and Forrest5 , Theorem 1·25]), it is connected (because both the base graph and the replacement graphs are connected, see [ Reference Belk and Forrest5 , Remark 1·27]) and it is locally connected (proven in Lemma 3·6 below). It contains no simple closed curve, as none of the expansions of $\mathcal{D}_n$ contain simple closed paths. Thus, the limit space of $\mathcal{D}_n$ is a dendrite (Definition 2·1), and since its points have order 1, 2 or n, it is homeomorphic to the Ważewski dendrite $D_n$ (Definition 2·2). We will thus use the same symbol $D_n$ to denote both the Ważewski dendrite and the limit space of $\mathcal{D}_n$ .

3·2. Branch points, branches, endpoints and arcs

In this Subsection we give a few definition that will be useful throughout this work.

Observe that each branch at p is the topological interior of the union of finitely many cells, one of which has boundary $\{ p \}$ . However, note that not every topological interior of the connected union of finitely many cells is a branch.

We will often refer to the following specific branch point:

\[ [\![ 11\bar{n} ]\!] = [\![ 21\bar{n} ]\!] = \dots = [\![ n1\bar{n} ]\!], \]

where the bar on top of n denotes periodicity. We call this point the central branch point and we denote by ${p_0}$ . There is nothing topologically special about this branch point, but its nice combinatorial description will make it handy in the remainder of this work.

Given a subset X of $D_n$ , we denote by $\mathbf{{Br}(X)}$ and $\mathbf{{En}(X)}$ the sets of all branch points and endpoints, respectively, that are contained in X. When $X = D_n$ itself, we simply write $\mathbf{{Br}}$ and $\mathbf{{En}}$ .

Note that every vertex of a graph expansion of $\mathcal{D}_n$ either corresponds to a branch point or an endpoint, depending on whether its degree is n or 1. Moreover, every branch point is a vertex of some graph expansion, but not every endpoint is a vertex, since En is uncountable. For concrete examples of endpoints that are not vertices, consider any sequence that does not contain 1 and that is not eventually repeating.

We say that an endpoint is a rational endpoint if it is a vertex of some graph expansion. We denote by $\mathbf{{REn}(X)}$ the set of those rational endpoints that are contained in $X \subseteq D_n$ , and we simply write $\mathbf{{REn}}$ when X is the entire dendrite $D_n$ . Note that, by the same reason mentioned in Remark 3·4, REn is dense in $D_n$ because every cell contains rational endpoints.

Being a rational endpoint is not a topological property: the set REn depends on the chosen homeomorphism between the limit space and the Ważewski dendrite $D_n$ , and in general a homeomorphism need not preserve the rationality of an endpoint. Since we have already fixed a homeomorphism between the limit space and the dendrite to begin with, there will be no need to distinguish between endpoints that are rational under a homeomorphism but not under another.

Differently from generic homeomorphisms, it is easy to see that an element of the rearrangement group $G_n$ must induce a permutation of REn.

On the other hand, every branch point of $D_n$ is rational, since it is a vertex of some graph expansion. In this sense, branch points are more useful when dealing with rearrangements, since the set of branch points is invariant under any homeomorphism.

Proof. Given a point $p \in D_n$ , we want to build a basis of connected open sets at p. We distinguish the case in which p corresponds to a vertex and the case in which it does not, and we will use the fact that each cell is connected (because the replacement graph is connected).

If p corresponds to a vertex of some graph expansion, then it is codified by a finite amount of sequences $x \alpha^{(1)}, \dots, x \alpha^{(m)}$ with a common prefix x (possibly empty) and m infinite sequences $\alpha^{(1)}, \dots, \alpha^{(m)}$ . More precisely, by Remark 3·5 if p is a branch point then $m=n$ and if p is a rational endpoint then $m=1$ . In both cases, a basis of connected open sets is given by the cells corresponding to the edges that are adjacent to p in the kth full expansion of $\mathcal{D}_n$ in the following way:

\[ \left\{ \{p\} \cup C^{\mathrm{o}}(x \alpha^{(1)}_1 \dots \alpha^{(1)}_k) \cup \dots \cup C^{\mathrm{o}}(x \alpha^{(m)}_1 \dots \alpha^{(m)}_k) \mid k \in \mathbb{N} \right\}. \]

Now consider a point p that is not a vertex. It corresponds to a unique infinite sequence $\omega = \omega_1 \omega_2 \dots$ , and must belong to the interior of every $C(\omega_1 \dots \omega_k)$ , so a basis of connected open sets is simply given by

\[ \{ C^{\mathrm{o}}(\omega_1 \dots \omega_k) \mid k \in \mathbb{N} \}. \]

We already mentioned arcs in Definition 2·2, describing them as homeomorphic copies of [0, 1]; naturally these two definitions are equivalent, but it is clear that, for dendrites, the existence and uniqueness of arcs for each pair of distinct points is worth remarking and plays a significant role in their topological properties and in the kind of homeomorphisms that are available on them. Indeed, this notion will be important for identifying the Thompson subgroups described later in Subsubsection 3·3·2, where we will consider the following special types of arcs.

Similarly to the rationality of an endpoint, the rationality of an arc is not a topological property, meaning that homeomorphisms need not preserve it. It is true, however, that a rearrangement must map a rational arc to a rational arc and a non-rational arc to a non-rational arc. In practice, we will only be working with arcs that are rational throughout the rest of this paper.

We will frequently use the following correspondence between paths in a graph expansion and rational arcs.

Remark 3·9. There is a natural surjective correspondence from the set of paths in graph expansions to the set of all rational arcs, which is the following. If $e = x_1 \dots x_k$ is an edge of some graph expansion, we associate to it the arc [p, q] for $p = [\![ x_1 \dots x_k 1 \bar{n} ]\!]$ and $q = [\![ x_1 \dots x_k \bar{n} ]\!]$ . This arc is precisely the set of points of $D_n$ that are represented by some sequence $x_1 \dots x_k \omega$ for $\omega$ in the alphabet $\{1,n\}$ . Inductively extending this construction gives the desired correspondence from paths of a graph expansions to rational arcs, which is surjective because clearly we can find a path between any two points of $D_n$ that are vertices. In short, if P is the path consisting of the edges $e_1, \dots, e_l$ (all of which are finite words), then the associated arc is

\[ A = \big\{ [\![ e_i \omega ]\!] \mid i=1, \dots, n \text{ and } \omega \text{ is a sequence in } \{1,n\} \big\}. \]

This correspondence is not injective, but the preimage of each rational arc contains a unique path with a minimal number of edges, which is obtained from any other edge of the same preimage by performing as many graph reductions as possible. When restricted to the set of these minimal paths, this correspondence is a bijection.

Remark 3·10. For each EE-arc A, there is a canonical homeomorphism between A and the unit interval [0, 1] that induces a bijection between the branch points of A and the dyadic points of (0, 1). The homeomorphism is built as described below using the path P from the previous Remark 3·9.

Given a single edge $e = x_1 \dots x_l$ , we consider the homeomorphism

\[ \Phi_e\,:\, \big\{ [\![ e \omega ]\!] \mid \omega \text{ is a sequence in } \{1,n\} \big\} \to \big[ 0, 2^{-l} \big] \]

that is built inductively by letting e 1 and e n correspond to $2^{-l} [0, 1/2]$ and $2^{-l} [1/2,1]$ in the natural order-preserving fashion, and so forth for further edge expansions.

Given a path P of edges $e_1, \dots, e_k$ , we build $\Phi_P$ as the concatenation of the $\Phi_{e_i}$ ’s, by which we mean that the first $\Phi_{e_1}$ maps to $[0, 2^{-l_1}]$ , the second $\Phi_{e_2}$ to the adjacent interval of length $2^{-l_2}$ (which is $[2^{-l_1}, 2^{-l_1}+2^{-l_2}]$ ) and so forth. For a generic path P, this covers some dyadic interval $[0, z] \subseteq [0, 1]$ , for some $z \in \mathbb{Z}[1/2]$ , and it is easy to see that $\Phi_P (A) = [0, 1]$ if and only if P corresponds to an EE-arc.

One may note that this construction is very similar to the usual homeomorphism between [0, 1] and the limit space of the replacement system for Thompson’s group F (depicted in Figure 5), which is simply given by identifying each number in [0, 1] with its binary expansion. For the replacement system of dendrite replacement systems (Figure 7), this is less straightforward due to the fact that the edge 1 is directed towards the initial vertex.

3·3. Fundamental subgroups of dendrite rearrangement groups

From here onward, given an $n \geq 3$ , we will denote by $G_n$ the rearrangement group of $\mathcal{D}_n$ , and we now begin its study.

In general, even if a replacement system produces an appealing limit space, its rearrangement group might be uninteresting or even trivial (see [ Reference Belk and Forrest5 , Example 2·5]). Here, however, we prove that every dendrite replacement system yields a rearrangement group that contains many copies of Thompson’s group F (Subsubsection 3·3·2), immediately hinting at the fact that these rearrangement groups are, in fact, quite large. How large they are will be made clear with Theorem 5·4, stating that each dendrite rearrangement group $G_n$ is dense in the full group of homeomorphisms of the Ważewski dendrite $D_n$ .

3·3·2. The thompson subgroup H

The second fundamental subgroup of $G_n$ is an isomorphic copy of Thompson’s group F. Let $A_0$ denote the EE-arc $[q_1, q_n]$ between the endpoints $q_1 = [\![ 1 \bar{n} ]\!]$ and $q_n = [\![ \bar{n} ]\!]$ . Note that, by Remark 3·9, $A_0$ corresponds to the set of those points of $D_n$ that are represented by some infinite sequence in the alphabet $\{ 1,n \}$ . The canonical homeomorphism from $A_0$ to [0, 1] described in Remark 3·10 prompts the following Lemma.

Lemma 3·13. For every $n \geq 3$ , the dendrite rearrangement group $G_n$ contains an isomorphic copy H of Thompson’s group F that acts on $A_0$ as F does on the unit interval. In particular, this subgroup H is generated by the two elements portrayed in Figure 8.

Proof. Consider the elements $g_0$ and $g_1$ depicted in Figure 8 (a precise definition in terms of sequences is given below) and let $H = \langle g_0, g_1 \rangle$ . A quick comparison with the standard generators $X_0$ and $X_1$ of Thompson’s group F (Figure 6) immediately shows that the subgroup H is isomorphic to Thompson’s group F.

It may be noted that this copy H of Thompson’s group F can also be produced by an application of [ Reference Belk and Forrest5 , Proposition 2·8]. In practice, up to ignoring a 180-degree flip (and, in general, the elements that switch the endpoints $[\![ 1\bar{n} ]\!]$ and $[\![ \bar{n} ]\!]$ ), the subgroup H can be thought of as the rearrangement group obtained by forgetting about the edges $2, 3, \dots, n-1$ in both the base and the replacement graphs.

Fig. 8. The two generators $g_0$ and $g_1$ of the copy H of Thompson’s group F (drawn with undirected edges, see Remark 3·1).

More detailed definitions of the generators $g_0$ and $g_1$ are the following (here we do not use undirected edges for the sake of completeness, so $g_0(C(11)) = C(n1)$ and $g_1(C(n1n)) = C(nn1)$ are expanded and follow the rules described in Remark 3·1):

where $\tilde{j}$ denotes $n+1-j$ and the i’s always represent all possible choices among $2, \dots, n-1$ .

In Section 4, when more about the transitivity properties of dendrite rearrangement groups will have been discussed, Lemma 4·5 will show that there are many more copies of Thompson’s group F nested inside $G_n$ .

Finally, we point out that actions of Thompson’s group F on dendrites have been considered in literature. For example, in [ 22 , Corollary 1·5] it is proved that such actions have an orbit that consists of one or two points, and in [ Reference Abdalaoui and Naghmouchi1 , Corollary 4·5] it is shown that such actions are equicontinuous on their minimal sets. The dissertation [ 46 ] is also about a Thompson-like group acting on a dendrite (namely, the Julia set for the complex map $z \to z^2 +i$ , which is homeomorphic to the Ważewski dendrite $D_3$ ): differently from our dendrite rearrangement groups, the group studied in [ 46 ] is much more T-like, as it is defined as a group of those homeomorphic of the unit circle that preserve a lamination induced by the Julia set. It seems likely that this group belongs to the family of orientation-preserving dendrite rearrangement groups defined in Subsection 8·2.

4·1. Some transitivity properties

For any $k \in \mathbb{N}$ , let $\mathbf{{Br}^{(k)}(X)}$ be the set of all subsets of $\mathrm{Br}(X)$ consisting of exactly k elements. From Lemma 3·13, it is clear that the action of H on the branch points of $A_0^{\mathrm{o}}$ (the interior of the arc $A_0$ ) corresponds to the action of F on the dyadic points of (0, 1). In particular, since it is known that for every $k \in \mathbb{N}$ Thompson’s group F acts transitively on the set of k-tuples of dyadic points of (0, 1), we have the following.

Using this Lemma, we can prove the following useful Proposition.

4·2. A finite generating set for $G_{n}$

The overall strategy of the proof of the following result is similar to that of [ Reference Tarocchi49 , Theorem 6·2], which is about the generating set of the airplane rearrangement group $T_A$ . The similarity is no coincidence: the close relation between the airplane rearrangement group and dendrite rearrangement groups will be discussed in Subsection 8·3.

Fig. 9. A schematic depiction of the element $h_D$ for the proof of Theorem 4·3, with $m=3$ . Here, $w_i$ denotes the image of $v_i$ .

4·3. Further transitivity properties

Let A be any rational arc (Definition 3·8) and consider the set $\mathrm{Br}(A^{\mathrm{o}})$ of branch points that are contained in the interior of A. The next Lemma shows that $G_n$ contains a copy of Thompson’s group F for every arc $A = [p,q]$ , where $p, q \in \mathrm{Br} \cup \mathrm{REn}$ .

Proof. It suffices to prove the statement for an EE-arc A. Indeed, each BE-arc or BB-arc is contained in an EE-arc and corresponds to a dyadic interval under the identification of A with [0, 1] described in Remark 3·10, and it is known that F contains a copy of itself on every dyadic interval that is contained in [0, 1].

Let $A = [p,q]$ be an EE-arc. As explained in Remark 3·9, consider the minimal expansion E of $\mathcal{D}_n$ that contains the rational endpoints p and q as vertices. In this graph, the arc A corresponds to a path P in the sense that, if $e_1, \dots, e_k$ are the edges of P, then

\[ A = \big\{ [\![ e_i \omega ]\!] \in D_n \mid i = 1, \dots, n \text{ and } \omega \text{ is a sequence in } \{ 1, n \} \big\}. \]

By induction on the length k of the path P in the minimal graph where both p and q appear, we will find an element $g \in G_n$ such that g(P) corresponds to $A_0$ in the sense described above, so that $g(A) = A_0$ . The goal will essentially be to find an element that modifies P so that a reduction occurs, in order to decrease k.

To make this easier, we will suppose that P contains the central branch point $p_0$ . We can do this without loss of generality up to conjugating by an element of $G_n$ that maps some branch point of P to $p_0$ , which exists thanks to Proposition 4·2.

The base case is $k=2$ , meaning that P consists of two edges of the base graph. In this case, the two edges correspond simply to distinct 1-letter words i and j chosen among $1, \dots, n$ . Then g is the unique element of K that switches i with 1 and j with n, leaving everything else unchanged.

For $k \geq 3$ , either p or q is not among the endpoints $[\![ i \bar{n} ]\!]$ , and without loss of generality we will assume that it is p. Denote by r the vertex of P that is adjacent to p. Observe that r must be a branch point, so $r = [\![ x i 1 \bar{n} ]\!]$ for some x that does not end with 1 nor with n (Remark 3·5). Then it is not possible that p is $[\![ x \bar{n} ]\!]$ , otherwise E would not be the minimal graph expansion where p and q appear, because the cell C(x) could be reduced (Figure 10 shows why this is the case with an example). We now find an element $g \in G_n$ that causes precisely this reduction, i.e., g is such that g(P) has one edge less than P in the minimal graph expansion where both g(p) and $g(q) = q$ appear (note that $g(q) = q$ because, as assumed at the beginning of the proof, P includes the central branch point $p_0$ , so q is not included in any of the two branches permuted by g). A simple example is depicted in Figure 11; more generally, in reference to this figure, there may be more vertices on the path joining r with g(p), in which case the edge between r and p needs to be expanded accordingly to find a graph pair diagram for g that switches p and g(p). Since $r = [\![ x i 1 \bar{n} ]\!]$ , we have that $p = [\![ x j \bar{n} ]\!]$ for some j that is not n (as noted above) nor 1 (or p would not be an endpoint). Then we choose g to be the unique involution that switches the edges xj and xn without changing anything else. Now, g(P) admits the aforementioned reduction and we can apply our induction hypothesis on g(p) and $g(q) = q$ , finding an element $h \in G_n$ such that $hg(A) = A_0$ and $H^{hg}$ is the desired copy of Thompson’s group F acting on A.

5·1. Transitivity properties of dendrite rearrangement groups

The next proposition is an analog of [ 23 , Proposition 6·1], which goes to show that $G_n$ has many of the transitivity properties of $\mathbb{H}_n$ despite being only a countable group. The construction of the tree $T(\mathcal{F})$ described below is heavily inspired by the one given at the beginning of [ 23 , section 6], which in turn is not much different from the construction used in [ Reference Belk and Forrest5 , Theorem 4·14] for studying the rearrangement group of the Vicsek fractal (which is homeomorphic to a dendrite, see Subsection 8·1). Similar tree approximations for dendrites are common and have appeared elsewhere in literature (see, for example, [ Reference S. B.37 , section X·3]).

Let $\mathcal{F}$ be a finite subset of Br and consider the unique subdendrite $[\mathcal{F}]$ of $D_n$ that contains $\mathcal{F}$ and is minimal with respect to set inclusion. It is homeomorphic to the geometric realisation of a finite simplicial tree that has a vertex for each element of $\mathcal{F}$ (and possibly more vertices where the tree needs to branch out). This simplicial tree is not unique, since we can choose to add vertices in the middle of any edge, so we choose $T(\mathcal{F})$ to be the minimal among these trees, and since $\mathcal{F} \subset \mathrm{Br}$ we have that each vertex of $T(\mathcal{F})$ is a branch point. A simple example is depicted in Figure 12.

Fig. 12. An example of the tree $T(\mathcal{F})$ depicted beside the finite subset $F \subset \mathrm{Br}$ , for $n=4$ .

This transitivity property of $G_n$ on the set of branch points is the same that the full homeomorphism group $\mathbb{H}_n$ has by [ 23 , Proposition 6·1], so Proposition 5·1 is essentially saying that every dendrite rearrangement group is as transitive as a subgroup of $\mathbb{H}_n$ can be on the set of branch points.

As an immediate consequence of Proposition 5·1, we have the two following facts (which are analogous to [ 23 , corollaries 6·5 and 6·7], respectively).

Recall that a G-action on a set X is oligomorphic if, for every $m \in \mathbb{N}$ , the diagonal G-action on the set of m-tuples of X has finitely many orbits. For more information about oligomorphic group actions, see [ Reference Cameron14 ].

5·2. Density in the full group of homeomorphisms

Here we prove that Proposition 5·1 has the following consequence.

As done in [ 23 ], here we endow the group $\mathbb{H}_n$ with its topology of uniform convergence, which makes it a Polish group (i.e., a separable and completely metrisable topological group). By [ 23 , Proposition 2·4], this topology of $\mathbb{H}_n$ is inherited by the pointwise convergence topology of $\mathrm{Sym}\big(\mathrm{Br}\big)$ , so in order to prove Theorem 5·4 it suffices to show that the permutations of Br induced by elements of $\mathbb{H}_n$ can be “approximated” by those of $G_n$ . Thus, Theorem 5·4 is equivalent to the following.

In the next Section, with Proposition 6·3 and Corollary 6·4, using these very same arguments, we will see that the commutator subgroup $[G_n, G_n]$ shares strong transitivity properties with $G_n$ and thus it is also dense in $\mathbb{H}_n$ despite being significantly “smaller” than $G_n$ (it has infinite index by Theorem 6·1).

Observe that being dense in an ambient group is not new behaviour in the world of Thompson groups. Indeed, [ 9 , Corollary A5·8] and [ Reference Zhuang52 , Proposition 4·3] show that Thompson groups F and T are dense in the groups $\mathrm{Homeo}^+([0, 1])$ and $\mathrm{Homeo}^+(S^1)$ of orientation-preserving homeomorphisms of the unit interval and the unit circle, respectively. Also, Thompson’s group V is dense in both the group $\mathrm{AAut}(\mathcal{T}_2)$ of almost automorphisms of the binary tree (noted in [ 16 ]) and also in the group $\mathrm{Homeo}(\mathfrak{C})$ of homeomorphisms of the Cantor space with its Polish compact-open topology (see Remark 5·7 below). Notably, the very recent work [ Reference Neretin39 ] shows that the rearrangement group of the Basilica is dense in the group of all orientation-preserving homeomorphisms of the Basilica. Later in Subsection 8·3 we will see that the Airplane rearrangement group is also dense in the group of orientation-preserving homeomorphisms of a dendrite. All of these results naturally prompt the following question, which was also asked in [ Reference Neretin39 , Final remarks (b)].

This question might be related to recent findings and questions about quasi-symmetries of fractals. Namely, [ 35 ] shows that the Basilica rearrangement group $T_B$ is in a certain sense dense in the group of planar quasi-symmetries of the Basilica Julia set, and [ Reference Belk and Forrest4 ] asks if it is always possible to find a finitely generated dense subgroup of the orientation-preserving group of homeomorphisms.

We believe that this question is interesting for practical reasons. Finding a Thompson-like group G that is dense in an interesting uncountable group $\mathbb{H}$ essentially allows to approximate the “non-computable” group $\mathbb{H}$ with the simpler group G that can probably be easily handled by a computer.

6·1. The abelianisation of dendrite rearrangement groups

In this Subsection we describe two families of maps: the local parity maps $\pi_p$ (p a branch point) and the local endpoint derivatives $\partial_q$ (q an endpoint), both satisfying a chain rule that will allow us to define global versions of these maps. Together, the parity map and the endpoint derivative will allow us to describe the elements of the commutator subgroup $[G_n, G_n]$ of a dendrite rearrangement group and to compute the abelianisation of $G_n$ by applying Schreier’s Lemma (which was also used in [ Reference Belk and Forrest3 ] for the computing abelianisation of the Basilica rearrangement group $T_B$ ).

6·1·1. The parity map

The branches at a given branch point p are naturally enumerated by $1, \dots, n$ . More precisely, there is a unique finite word x for which $p = [\![ x i 1 \overline{n} ]\!]$ ; each branch at p intersects precisely one of the cells $[\![ x i ]\!]$ (and thus every cell whose address starts with xi): this induces an enumeration of the branches at p (i.e., a bijection with $\{1, \dots, n\}$ ).

A homeomorphism g of $D_n$ induces a bijection between the branches at p and those at g(p). This in turn induces a permutation $\sigma_{p,g} \in \mathrm{Sym}(n)$ according to the enumeration of the branches at p and g(p). We define the local parity map at the branch point $p \in \mathrm{Br}$ as the mapping

\[ \pi_p \colon G_n \to \mathbb{Z}_2 \]

such that $\pi_p(g) = 0$ if $\sigma_{p,g}$ is an even permutation and $\pi_p(g) = 1$ if $\sigma_{p,g}$ is an odd permutation.

Note that the local parity maps satisfy the following chain rule:

\[ \pi_p (g h) = \pi_{h(p)} (g) + \pi_p (h). \]

Observe that, whenever a branch point p does not appear in a graph pair diagram of a rearrangement g, one has $\pi_p(g)=0$ because canonical homeomorphisms of a cell C(x) (which were defined at the beginning of Subsection 2·4) preserve the enumeration of branches around the branch point $[\![ x i 1 \bar{n} ]\!]$ . In particular, this implies that the definition does not depend on the choice of graph pair diagram chosen to represent g. This, together with the previously described chain rule and the fact that every rearrangement induces a permutation of Br, allow us to define the (global) parity map $\Pi$ as

\[ \Pi \colon G_n \to \mathbb{Z}_2, \; g \mapsto \sum_{p \in \mathrm{Br}} \pi_p(g). \]

Because of the chain rule identity, it is easy to see that $\Pi$ is a group morphism.

6·1·2. The endpoint derivative

We define the local endpoint derivative at the rational endpoint $q \in \mathrm{REn}$ as the mapping

\[ \partial_q \colon G_n \to \mathbb{Z} \]

defined as follows. If (D, R, f) is a graph pair diagram for g, there is a unique edge x of D that is incident on q, and a unique edge $y = f(x)$ of R that is incident on f(q). Denote by $L_D(x)$ and $L_R(y)$ the number of n’s at the end of x and y, without considering the first letter in case the word is $n n \dots n$ (the first letter is special in the sense that it denotes an edge of the base graph instead of the replacement graph). Then

\[ \partial_q(g) = L_D(x) - L_R(y). \]

This definition does not depend on the choice of graph pair diagram chosen to represent g, since expanding x in D always causes the expansion of y in R, and both the resulting lengths $L_D$ and $L_R$ increase by 1, leaving $\partial_q$ ultimately unchanged.

The intuition behind this definition is that $L_D(x)$ (or $L_R(y)$ ) represent how small of a portion of a “straight branch” x (or y) is, when drawing z1 and zn straight and aligned with z when expanding an edge z. In this sense, $\partial_q$ looks like a logarithmic derivative in a small enough neighbourhood of the endpoint q.

Note that, as the local parity maps do, the endpoint derivatives also satisfy the following chain rule:

\[ \partial_q (g h) = \partial_{h(q)} (g) + \partial_q (h). \]

Observe that, whenever a rational endpoint q does not appear in a graph pair diagram of a rearrangement g, one has $\partial_q(g)=0$ . Then, for the same reasons used for the parity map, we can define the (global) endpoint derivative $\Delta$ as

\[ \Delta \colon G_n \to \mathbb{Z}, \; g \mapsto \sum_{q \in \mathrm{REn}} \partial_q(g). \]

Because of the chain rule identity, $\Delta$ is a group morphism.

Before proceeding with the computation of the abelianisation of $G_n$ , we would like to mention that this idea of an endpoint derivative was previously used to characterize the commutator subgroup $[T_A, T_A]$ of the Airplane rearrangement group. This in turn is related to the fact that the commutator subgroup of Thompson’s group F consists of the elements of F that act trivially around the neighbourhoods, i.e., that preserve the numbers of 0’s (respectively 1’s) of the leftmost (respectively rightmost) edge of a graph pair diagram. There was no parity map for $T_A$ essentially because the subgroup of $T_A$ that corresponds to $K \leq D_n$ is Thompson’s group T (see Subsection 8·3), which is simple.

6·1·3. Computing the abelianisation

Consider the following map:

\[ \Phi = \Pi \times \Delta \colon G_n \to \mathbb{Z}_2 \oplus \mathbb{Z}, \; g \mapsto \big( \Pi(g), \Delta(g) \big). \]

Clearly $\Phi$ is a group morphism, and it is not hard to see how it behaves on the generators $\{g_0, g_1, \tau_2, \dots, \tau_n \}$ of $G_n$ , where each $\tau_i$ is the element of K that corresponds to the permutation (1 i) and $g_0$ and $g_1$ were represented in Figure 8 and generate H.

\begin{align*} &\Phi(g_0) = (0,0), \; \Phi(g_1) = (0,-1),\\ &\,\,\,\Phi(\tau_i) = (1,0), \; \forall i = 2, \dots, n.\end{align*}

The parity map on $g_0$ and $g_1$ is 0 because both of these elements feature exactly two branch points with the same non-trivial permutation (for $g_0$ these are $[\![ i \overline{n} ]\!]$ and $[\![ 1 1 i \overline{n} ]\!]$ while the permutation is trivial at $[\![ 1 i \overline{n} ]\!]$ , while for $g_1$ they are $[\![ n i \overline{n} ]\!]$ and $[\![ n 1 1 i \overline{n} ]\!]$ while the permutation is trivial at $[\![ i \overline{n} ]\!]$ and $[\![ n i \overline{n} ]\!]$ ). In particular, from these computations we deduce that $\Phi$ is surjective, because for all $\zeta \in \{0, 1\}$ and $z \in \mathbb{Z}$ one has $\Phi(\tau_2^\zeta g_1^z) = (\zeta, -z)$ . If we let $K = \mathrm{Ker}(\Phi)$ , then $G_n / K \simeq \mathbb{Z}_2 \oplus \mathbb{Z}$ , so $[G_n, G_n] \leq K$ , so our goal is to show that $K \leq [G_n, G_n]$ .

Using the following set of transversals for the quotient $G_n / K$

\[ \left\{ \tau_2^{\zeta} g_1^{-z} \mid \zeta \in \{0, 1\}, z \in \mathbb{Z} \right\} \]

and the generating set $\{g_0, g_1, \tau_2 \dots, \tau_n \}$ for $G_n$ (Theorem 4·3), together with the application of the identities $\tau_2 g_1 = g_1 \tau_2$ and $\tau_i = \tau_i^{-1}$ , an application of Schreier’s Lemma (see, for example, [ 43 , Lemma 4·2·1]) produces the following generating set for K:

(6·1) \begin{equation} \left\{ g_1^{-z} \tau_2 \tau_i g_1^z, \; \tau_2^\zeta g_1^{-z} g_0 g_1^z \tau_2^\zeta \, \mid \, \zeta \in \{0, 1\}, z \in \mathbb{Z}, i = 3, \dots, n \right\}.\end{equation}

This generating set is included in the normal closure of $\{ \tau_2 \tau_i, g_0 \mid i = 3, \dots, n \}$ in $G_n$ and the commutator subgroup is normal in $G_n$ , so it suffices to show that $[G_n, G_n]$ contains $\tau_2 \tau_3, \dots, \tau_2 \tau_n$ and $g_0$ in order to be able to conclude that $K \leq [G_n, G_n]$ . For each $i \in \{3, \dots, n\}$ the element $\tau_2 \tau_i$ is an even permutation, so it belongs to $[K, K] \leq [G_n, G_n]$ . As for $g_0$ , using the fact that $[H,H] \simeq [F,F]$ is the subgroup of the elements of H that act trivially around the endpoints of C(1) and C(n), a direct computation shows that

(6·2) \begin{equation}g_0 = h [\tau_n, g_1] \text{ for an element } h \in [H,H],\end{equation}

which proves that $g_0 \in [G_n, G_n]$ and ultimately that $[G_n, G_n]$ is the kernel of $\Phi$ .

To sum up the results shown so far about the commutator subgroup:

Theorem 6·1. The commutator subgroup $[G_n, G_n]$ of a dendrite rearrangement group $G_n$ is the kernel of the surjective morphism $\Phi = \Pi \times \Delta\,:\, G_n \to \mathbb{Z}_2 \oplus \mathbb{Z}$ . In particular, the abelianisation of $G_n$ is $\mathbb{Z}_2 \oplus \mathbb{Z}$ . Moreover, $[G_n, G_n]$ is generated by the (infinite) set exhibited above in (6·1).

Remark 6·2. Using the identities $\tau_i g_1 = g_1 \tau_i$ for $i = 2, \dots, n-1$ and $[\tau_n, g_1^z] = [\tau_n, g_1]^z$ (which can both be verified with quick computations), the part $g_1^{-z} \tau_2 \tau_i g_1^z$ of the generating set in (6·1) can be replaced by the finite subset

\[ \{ \tau_2 \tau_i, [\tau_n, g_1] \mid i = 3, \dots, n \}. \]

This produces the following generating set for $[G_n, G_n]$ , which will be useful at the end of the proof of Theorem 6·6:

\[ \{ \tau_2 \tau_i, \; [\tau_n, g_1], \; \tau_2^\zeta g_1^{-z} g_0 g_1^z \tau_2^\zeta \, \mid \, \zeta \in \{0, 1\}, z \in \mathbb{Z}, i = 3, \dots, n \}. \]

This infinite generating set will be refined to a finite generating set in Theorem 6·9 at the end of this Section.

6·2. Simplicity of the commutator subgroup

An analogue of Proposition 5·1 also applies to the commutator subgroup $[G_n, G_n]$ . In order to prove this, we need the two following notions. If a group G acts on a space X, the support of an element $g \in G$ is the set of points that are not fixed by g. If $U \subseteq X$ then the rigid stabiliser of U in G is the subgroup of those elements whose support is included in U, i.e., the elements that act trivially outside of U. In symbols, this is

\[ \boldsymbol{Rist}_{\boldsymbol{G}}\boldsymbol{U} = \{ g \in G \mid g(x) = x, \, \forall x \in X \setminus U \} \leq G. \]

The notion of rigid stabilisers will also be essential in Theorem 6·6.

Using the exact same argument of Claim 5·5, this Proposition allows to immediately prove that the commutator subgroup $[G_n, G_n]$ is also dense in the full group of homeomorphisms of $D_n$ .

As another consequence of Proposition 6·3, we have:

This last Corollary allows to prove the simplicity of $[G_n, G_n]$ for all $n \geq 4$ .

Proof. This proof shares its structure with the proofs of the simplicity of the commutator subgroups of the Basilica and Airplane limit spaces (from [ Reference Belk and Forrest3 ] and [ Reference Tarocchi49 ], respectively), which in turn were inspired by the so-called Epstein’s double commutator trick from [ Reference Epstein25 ].

Given a non-trivial normal subgroup N of $[G_n, G_n]$ , we wish to prove that $N = [G_n, G_n]$ . Let $g \in N \setminus \{1\}$ . Since g is non-trivial, there must be an open subset U of $D_n$ such that $U \cap g(U) = \emptyset$ . Every open subset of $D_n$ contains a cell (see [ 41 , Lemma 2·2]) and every cell contains a branch; thus, there exists a branch B based at some branch point p that is included in U, and so $B \cap g(B) = \emptyset$ .

Let $h_1, h_2 \in \mathrm{Rist}_{[G_n, G_n]}(B)$ . Note that $g^{-1} h_1 g \in \mathrm{Rist}_{[G_n, G_n]} \big( g^{-1}(B) \big)$ , so

\[ [h_1, g] = h_1^{-1} g^{-1} h_1 g \in \mathrm{Rist}_{[G_n, G_n]} \big( B \cup g^{-1}(B) \big) \]

and $[h_1, g] |_B = h_1^{-1} |_B$ . Thus, since $h_2 \in \mathrm{Rist}_{[G_n, G_n]}(B)$ , direct computations of $\big[ [h_1, g], h_2 \big]$ restricted on B, $g^{-1}(B)$ and elsewhere show that globally

\[ \big[ [h_1, g]^{-1}, h_2 \big] =[h_1,g] h_2^{-1} [h_1,g]^{-1} h_2 = h_1^{-1} h_2^{-1} h_1 h_2 = [h_1, h_2]. \]

Now, since $[h_1, g] \in N$ (because $g \in N \trianglelefteq [G_n, G_n] \ni h_1$ ) and $h_2 \in [G_n, G_n]$ , the double commutator of the previous identity belongs to N, and so we have that

\[ [h_1, h_2] \in N \text{ for all } h_1, h_2 \in \mathrm{Rist}_{[G_n, G_n]}(B). \]

We do not have much control over the choice of B, but this can be overcome with the aid of Corollary 6·5: for any branch $B^*$ , there exists some $h \in [G_n, G_n]$ such that $h(B^*) = B$ , and conjugating the previous identity by h yields

(6·3) \begin{equation}[h_1, h_2] \in N \text{ for all } h_1, h_2 \in \mathrm{Rist}_{[G_n, G_n]}(B^*).\end{equation}

As is shown below, this identity allows us to show that N contains the generating set for $[G_n, G_n]$ exhibited in Remark 6·2, which will conclude this proof.

Let us first show that $\tau_2 \tau_i \in N$ for all $i = 3, \dots, n$ because N contains the entire subgroup [K, K]. Each 3-cycle (ijk) can be written as [(ik),(jk)]. Since (ik) and (jk) do not belong to $[G_n,G_n]$ , we need to replace them by some $h_1 = (ik)h$ and $h_2=(jk)h$ in $\mathrm{Rist}_{[G_n,G_n]}(B^*)$ (for any branch $B^*$ ) such that $(ijk) = [(ik), (jk)] = [h_1,h_2]$ , in order to apply (6·3). Using the fact that $n\geq4$ , such an h can be found, for example, as follows. Let $p' = [\![ kl1\overline{n} ]\!]$ (for any $l=1,\dots,n$ ), which is the central branch point of the cell C(k). Let h be the permutation at p’ that only switches the cells $C(k (n-1))$ and C(k n), which is an odd permutation whose support is included in $C^{\mathrm{o}}(k)$ . Then $h_1 = (ik)h$ and $h_2 = (jk)h$ are compositions of odd permutations, thus $h_1, h_2 \in [G_n,G_n]$ . Since $n\geq4$ , both $h_1$ and $h_2$ act trivially on some branch $\tilde{B}$ at $p_0$ that is different from i, j and k. (This last sentence does not work when $n=3$ , see Remark 6·7 and Question 6·8 below.) If we fix any branch point $p^*$ of $\tilde{B}$ , the elements $h_1$ and $h_2$ belong to $\mathrm{Rist}_{[G_n,G_n]}(B^*)$ , where $B^*$ is the unique branch at $p^*$ that includes $p_0$ . Thus, one concludes by (6·3).

To show that $[\tau_n, g_1]$ belongs to N, we need to rewrite it as $[\tau_n \rho, g_1 f]$ for suitable elements $\rho$ and f so that we can apply (6·3). To find $\rho$ and f such that $\rho\tau_n$ and $g_1f$ belong to $[G_n,G_n]$ , we can employ a strategy similar to that used in Proposition 6·3. For example, let $\rho$ be any odd permutation with support included in C(212) (for instance, the element that switches C(2122) and C(2123)) and let f be the element that acts on C(2n) as $g_1$ doe on C(n). Then $\tau_n\rho$ and $g_1f$ belong to $\mathrm{Ker}(\Phi)=[G_n,G_n]$ . Moreover, the supports of $\rho$ and f are disjoint and each of them has trivial intersection with the supports of $\tau_n$ and $g_1$ , so $[\tau_n, g_1] = [\tau_n \rho, g_1 f]$ . Since the union U of the supports of $\rho$ , f, $\tau_n$ and $g_1$ is a proper closed subset of $D_n$ , it is included in some branch B. It follows that both $\tau_n \rho$ and $g_1 f$ belong to $\mathrm{Rist}_{[G_n, G_n]}(B)$ , so we can conclude that $[\tau_n, g_1] = [\tau_n \rho, g_1 f] \in N$ by (6·3).

Now that we know that N contains $[\tau_n, g_1]$ , because of (6·2) it suffices to prove that $[H,H] \leq N$ in order to show that $g_0 \in N$ (recall that H is the Thompson subgroup described in Subsubsection 3·3·2). This is done by recalling that the commutator subgroup of Thompson’s group F is the subgroup of F that acts trivially around the endpoints of the unit interval, and that it is simple and non-abelian, so the commutator of [H, H] is [H, H] itself. Then an arbitrary element of [H, H] can be written as the product of commutators of elements of [H, H], each of which must be acting trivially on some neighborhood of the endpoints of C(1) and C(n). Then these elements act trivially in some small enough branch B and an application of (6·3) allows us to conclude that $[H,H] \leq N$ , and so that $g_0 \in N$ .

Finally, (6·2) tells us that $g_0 = h [\tau_n, g_1] = [f_1, f_2] [\tau_n, g_1]$ for some $f_1, f_2 \in [H,H]$ (using again the fact that the commutator subgroup of [H, H] is [H, H] itself). With this identity, the remaining elements of the generating set from Remark 6·2 all belong to N because they are

\[ \tau_2^\zeta g_1^{-z} g_0 g_1^z \tau_2^\zeta = g_0^{g_1^z\tau_2^{\zeta}} = \big[ f_1^{g_1^z\tau_2^{\zeta}}, f_2^{g_1^z\tau_2^{\zeta}} \big] \big[ \tau_n^{g_1^z\tau_2^{\zeta}}, g_1^{g_1^z\tau_2^{\zeta}} \big], \]

where each of the two commutators satisfy (6·3).

This question is left unanswered:

6·3. A finite generating set for the commutator subgroup

As a final result about the commutator subgroup of $G_n$ , we construct a finite generating set. The proof follows the same strategy of [ Reference Tarocchi49 , Theorem 7·12] and also makes use of the fact that the commutator subgroup of a Thompson subgroup $H_A$ is the set of elements of $H_A$ that act trivially on some neighborhood of each of the two endpoints of the arc A.

Let $q_0 = [\![ \overline{n} ]\!]$ and $A_\star = [p_0,q_0]$ (which is the “right arc” attached to $p_0$ ) and denote by $q_1$ the point $[\![ n i 1 \overline{n} ]\!]$ (which is the “middle point” of $A_\star$ ). We start from the generating set from Remark 6·2, and we will show that those elements all belong to the subgroup

\[ G_\star = \langle \tau_2 \tau_3, \dots, \tau_2 \tau_n, [\tau_n, g_1], g_0, g_0^{\tau_2}, g_\star \rangle, \]

where $g_\star$ is the element of [H, H] that acts on the arc $[p_0,q_1]$ (the left half of the right arc $A_\star = [p_0, q]$ ) as $g_0$ does on the entire arc $A_0$ . By definition $G_\star$ contains $\tau_2 \tau_i$ for each $i = 1, \dots, n$ and $[\tau_n, g_1]$ , so it suffices to show that the elements $\tau_2^\zeta g_1^{-z} g_0 g_1^z \tau_2^\zeta$ all belong to $G_\star$ .

For $\zeta = 0$ , observe that every element $g_1^{-z} g_0 g_1^z$ has the shape sketched in Figure 13, from which we deduce that it belongs to $[H,H] g_0$ . Now, we can prove that $[H,H] \leq G_\star$ with the following strategy: if $h \in [H,H]$ , then $h^{g_0^k}$ belongs to the commutator subgroup of $H_{A_\star}$ for a negative integer k that is small enough. But $[H_{A_\star}, H_{A_\star}]$ is included in the commutator subgroup of $\langle [\tau_n, g_1], g_\star \rangle$ , as the two elements act on $A_\star$ as the two generators of F do on [0, 1] (this step is discussed in more detail in [ Reference Tarocchi49 , Theorem 7·12]). Since $[\tau_n,g_1], g_\star \in G_\star$ , we have that $[H_{A_\star}, H_{A_\star}]$ is included in $G_\star$ . Thus, $[H,H] \leq G_\star$ , so $g_1^{-z} g_0 g_1^z \in G_\star$ for all $z \in \mathbb{Z}$ .

Fig. 13. The general shape of the elements $g_1^{-z} g_0 g_1^z$ , with an action on the curvy lines that depends on z.

As for the case $\zeta = 1$ , consider the arc $A_0' = [t,q]$ , where $t = [\![ 2 \overline{n} ]\!]$ and, as before, $q = [\![ ni1\overline{n} ]\!]$ . One can prove that the elements $\tau_2 g_1^{-z} g_0 g_1^z \tau_2$ (similar to Figure 13, but replacing the left central branch with the top one) all belong to $G_\star$ essentially by replacing $g_0$ with $g_0^{\tau_2}$ (that acts on $A_0'$ as $g_0$ does on $A_0$ ) and H with $H_{A_0'}$ in the previous paragraph. This simply means switching the first and second branches at $p_0$ , which preserves the arc $A_\star$ that we used as a pivot, so one “pushes” the action of $[H_{A_0'}, H_{A_0'}]$ inside $A_0$ by conjugating by negative powers of $g_0^{\tau_2}$ and obtains that $[H_{A_0'}, H_{A_0'}] \leq G_\star$ .

Ultimately, this shows that $G_\star \geq [G_n, G_n]$ . The converse is clear, so we ultimately have the following.

7·1. Finite subgroups

Generalising the ideas used in [ Reference Belk and Forrest5 , Proposition 2·10] (which is about the Vicsek rearrangement group, see Subsection 8·1), we obtain the following result.

Proposition 7·1. Each finite subgroup of $G_n$ is a subgroup of the group of automorphisms of some finite n-regular tree, and all such subgroups for all finite n-regular trees are achieved.

In particular, each finite subgroup of $G_n$ has order

\[ p_0^{\varepsilon} p_1^{k_1} \dots p_m^{k_m}, \]

where $p_1, \dots, p_m$ are the prime divisors of $(n-1)!$ , $k_i \in \mathbb{N}$ (possibly zero), $\varepsilon \in \{0, 1\}$ and $p_0 = n$ if it is prime and it is 1 otherwise. Every such order is achieved by some abelian subgroup.

Proof. First, let G be a finite subgroup of $G_n$ . It is shown in [ Reference Belk and Forrest5 , Theorem 2·9] that every finite subgroup of a rearrangement group is the subgroup of the automorphism group of some graph expansion of the replacement system, so $G \leq \mathrm{Aut}(T)$ for some finite n-regular tree T. It is easy to see that every finite n-regular tree can be realised as a graph expansion of $\mathcal{D}_n$ .

The decomposition of $\mathrm{Aut}(T)$ described in this paragraph is well-known in literature and is often called wreath recursion (see for example [ 38 , Section 1·4]). It is known that every finite tree has a unique center, which is either a vertex or an edge that is fixed by every automorphism (see for example [ Reference Meier36 , Theorem 3·46]). When an edge e is fixed, it is still possible that $\iota(e)$ and $\tau(e)$ are switched. In any case, $\mathrm{Aut}(T)$ can be decomposed as a finite sequence of semidirect products (where the action is by permutations of the components of the direct product) in the following way:

\[ \mathrm{Aut}(T) = A_0 \ltimes (A_1 \times \dots \times A_n), \]

where $A_0 \leq \mathrm{Sym}(n)$ or $A_0 \leq \mathrm{Sym}(2)$ (depending on whether the center is a vertex or an edge) and each $A_i$ is

\[ A_i = A_{i,0} \ltimes (A_{i,1} \times \dots \times A_{i,n-1}), \]

where $A_{i,0} \leq \mathrm{Sym}(n-1)$ and each $A_{i,j}$ is built in this very same way, eventually ending with the trivial group. In particular, this implies that the order of $\mathrm{Aut}(T)$ (and of each of its subgroups) is of the type exhibited in the statement.

Now, given a number of the form exhibited in the statement, we can write it as a product $q_1 \dots q_k$ of primes $q_i$ ’s that need not be pairwise distinct. Without loss of generality, we order them so that $q_1 = n = p_0$ if it is prime and it is one of the factors. Consider a rooted tree $\mathfrak{T}$ whose root has $q_1$ children whose ith level vertices all have $q_{i+1}$ children (note in particular that each vertex has degree that is at most n, by the choice $q_1 = n$ if it is a prime factor). Denote the root by the empty word and each ith child of w by wi. The abelian group $C_{q_1} \times \dots \times C_{q_k}$ acts faithfully on this tree with the diagonal action defined as follows: an element $(z_1, \dots, z_k)$ maps each vertex $x_1 \dots x_k$ to $(x_1 + q_1) \dots (x_k + q_k)$ . Since each vertex of $\mathfrak{T}$ has degree that is at most n, this action can be extended to a faithful action on the natural completion of $\mathfrak{T}$ to an n-regular tree. In particular, this group embeds into the automorphism group of an n-regular tree, so we are done.

7·2. Pairwise distinction

Using Proposition 7·1 from the previous Subsection, one can immediately distinguish two dendrite rearrangement groups $G_n$ and $G_m$ using the orders of their finite subgroups whenever n or m is at most 4 (they can actually be distinguished in this way for all $n, m \leq 9$ ). For $n \geq 5$ , we can use the same argument that is mentioned in [ Reference Belk and Forrest5 , subsection 2·3] for distinguishing the Vicsek rearrangement groups. For the sake of completeness, here we explain this argument in full detail for dendrite rearrangement groups.

To prove this Claim, assume that $A_m \simeq H \leq \mathrm{Aut}(\mathcal{T})$ . As noted in the previous Subsection 7·1, H must fix a vertex or an edge of $\mathcal{T}$ (see for example [ Reference Meier36 , Theorem 3·46]). Up to a barycentric subdivision of the fixed edge, we can assume that H is fixing a vertex r, so we can think of $\mathcal{T}$ as if it were a rooted tree. Then observe that, for each $k \in \mathbb{N}$ , the subgroup consisting of the elements of H that act trivially on the first k levels of $\mathcal{T}$ is normal in H. Since H is simple, this implies that for every k either H acts trivially or faithfully on the first k levels. Thus, since H is not trivial, there exists a k such that the action of H on the first $k-1$ levels is trivial whereas the action on level k is faithful. Note that then H must embed into a direct product of symmetric groups, i.e.,

\[ H \leq \mathrm{Sym}(m_1) \times \dots \times \mathrm{Sym}(m_l), \]

where the $m_i$ ’s are the number of children of the l vertices at level k. We now prove that this cannot happen unless there exists a j such that $m_j \geq m$ . This will conclude the proof, since each $m_i$ is either the degree of a vertex of $\mathcal{T}$ or the degree minus 1, depending on whether $k=0$ or $k\gt0$ .

Denote by $\mathrm{pr}_i$ the ith projection in the aforementioned direct product. Since H is simple, each $\mathrm{pr}_i(H)$ is either trivial or isomorphic to $H \simeq A_m$ . These cannot all be trivial or H would be trivial itself, so there is a j such that $A_m \simeq \mathrm{pr}_j(H) \leq \mathrm{Sym}(m_j)$ . It is known that $\mathrm{Sym}(m_j)$ cannot contain a copy of $A_m$ unless $m \leq m_j$ , so we are done.

As an immediate consequence of Proposition 7·2, we have the following.

Before moving on to a comparison with other Thompson groups, we highlight the following fact.

7·3. Relations with other Thompson groups

As seen in Subsubsection 3·3·2, dendrite rearrangement groups feature infinitely many isomorphic copies of Thompson’s group F. Here we show how the $G_n$ ’s compare to the bigger siblings of F.

Since both the rearrangement groups of the Basilica and of the Airplane limit spaces (respectively $T_B$ from [ Reference Belk and Forrest3 ] and $T_A$ from [ Reference Belk and Forrest5 , Reference Tarocchi49 ]) contain copies of T and embed in T, we can also conclude the following.

It appears that the groups $T_B$ and $T_A$ , along with Thompson’s group T, have more in common with the orientation-preserving dendrite rearrangement groups mentioned in Subsection 8·2 below. Instead, dendrite rearrangement groups $G_n$ do not preserve an orientation of the limit space $D_n$ on which they act, which is arguably what separates these groups from $T_B$ and $T_A$ .

The only other rearrangement group mentioned in literature that is likely to contain a copy of or embed into some dendrite rearrangement group is the Vicsek rearrangement group, along with its generalisations (see [ Reference Belk and Forrest5 , Example 2·1]). See Subsection 8·1 below for more about this comparison.

7·4. A solution to the conjugacy problem

We say that the conjugacy problem of a group G is solvable if there exists an algorithm that, given two elements of G, infallibly decides in finite time whether those elements belong to the same conjugacy class.

The main result of [ Reference Tarocchi48 ] states that, given an expanding replacement system whose replacement rules are reduction-confluent, the conjugacy problem is solvable in the associated rearrangement group. The solution described in [ Reference Tarocchi48 ] was inspired by a solution to the conjugacy problem for Thompson groups F, T and V from [ Reference Belk and Matucci6 ] and makes use of strand diagrams. The following paragraphs succinctly explain what reduction-confluent means and why dendrite replacement systems satisfy such property.

A graph reductions is the opposite operation of graph expansions of the replacement system. More precisely, given a graph $\Gamma$ , let $\Lambda$ be a subgraph isomorphic to a replacement graph $R_c$ . Let $\phi$ be such an isomorphism and suppose that the image of every vertex of $R_c$ other than its initial and terminal vertices is not incident on vertices of $\Gamma \setminus \phi(R_c)$ (i.e., only the initial and terminal vertices are allowed to be interact with the rest of $\Gamma$ ). Then we replace $\phi(R_c)$ with an edge coloured by c with initial vertex $\phi(\iota)$ and terminal vertex $\phi(\tau)$ . This defines a graph rewriting system, whose objects are finite graphs (up to isomorphisms) and whose operation of reduction we denote by $\longrightarrow$ . When edges are undirected (Remark 3·1), we do not distinguish between their orientations for the purposes of determining graph isomorphisms and graph reductions.

A rewriting system is confluent if, whenever $A \dashrightarrow B_1$ and $A \dashrightarrow B_2$ are finite sequences of rewritings, there exist an object C and a finite (possibly empty) sequences of rewritings $B_1 \dashrightarrow C$ and $B_2 \dashrightarrow C$ (i.e., when rewriting the same object A in two different ways, one can always continue rewriting to a common object C). We say that a replacement system is reduction-confluent if the the rewriting system of its graph reductions is confluent.

Let us see why dendrite replacement systems $\mathcal{D}_n$ have confluent graph reductions. Since graph reductions decrease the number of edges, they are terminating (i.e., there is no infinite sequence of graph reductions). Then, by Newman’s diamond Lemma [ 40 ], it suffices to show that they are locally confluent, i.e., whenever a graph $\Gamma$ and two distinct reductions $\Gamma \longrightarrow \Gamma_1$ and $\Gamma \longrightarrow \Gamma_2$ , there exist a graph $\Gamma^*$ and a finite (possibly empty) sequences of reductions $\Gamma_1 \dashrightarrow \Gamma^*$ and $\Gamma_2 \dashrightarrow \Gamma^*$ . In the case of dendrites, a graph reduction is uniquely identified by a vertex of degree n and its n incident edges. Assume first that the two reductions $\Gamma \longrightarrow \Gamma_1$ and $\Gamma \longrightarrow \Gamma_2$ involve no common edges, such as those portrayed in Figure 14 (edges are drawn without orientation because they are undirected, see Remark 3·1). Then such reductions can be performed one after the other, whatever the order, meaning that $\Gamma \longrightarrow \Gamma_1 \longrightarrow \Gamma^*$ and $\Gamma \longrightarrow \Gamma_2 \longrightarrow \Gamma^*$ , as needed. Assume instead that the graphs involved in the two reductions $\Gamma \longrightarrow \Gamma_1$ and $\Gamma \longrightarrow \Gamma_2$ share some edge. Then they can only share one edge, otherwise they would be the same reduction. The subgraph involved in both reductions is thus the one represented in Figure 15, from which it is clear that $\Gamma_1$ and $\Gamma_2$ are isomorphic, so in this case we have $\Gamma \longrightarrow \Gamma_1 \simeq \Gamma^*$ and $\Gamma \longrightarrow \Gamma_2 \simeq\Gamma^*$ . Ultimately, dendrite replacement systems are reduction-confluent, so their conjugacy problem is solvable by [ Reference Tarocchi48 ].

Fig. 14. Two reductions of graphs that share no edge, for the replacement system $\mathcal{D}_4$ .

Fig. 15. Two reductions of graphs that share an edge, for the replacement system $\mathcal{D}_4$ .

We note here that conjugacy in the (topological) group $\mathrm{AAut}(\mathcal{T}_{d,k})$ of almost automorphism of trees has been studied recently in [ 29 ]. That paper shows that whether two hyperbolic elements of $\mathrm{AAut}(\mathcal{T}_{d,k})$ are conjugate can be established by studying elements of the Higman–Thompson group $V_{d,k}$ that “approximate” the two almost automorphisms. Since we have seen that dendrite rearrangement groups “approximate” the full groups of homeomorphisms (Theorem 5·4), inspired by the use that [ 29 ] made of the strand diagrams developed in [ Reference Belk, Elliott and Matucci7 ], it is natural to ask the following.

7·5 Absence of invariable generation

We say that a group G is invariably generated if it admits a generating set S that still generates G even after one has conjugated each of the elements of S by elements of G. More precisely:

\[ G = \langle s^{g_s} \mid s \in S \rangle \text{ for every choice of } \{g_s \mid s \in S \} \subseteq G. \]

It was proved in [ Reference Gelander, Golan and Juschenko27 ] that Thompson’s group F is invariably generated, whereas the bigger siblings T and V are not.

A rearrangement group is weakly cell-transitive if, given an arbitrary cell C and an arbitrary proper union K of finitely many cells, there exists a rearrangement that maps K inside C. It was shown in [ 41 , Proposition 3·3] that this property of a rearrangement group is equivalent to other transitivity properties that can be found in literature, namely CO-transitivty and flexibility.

The main result of [ 41 ] is that, if a rearrangement group is weakly cell-transitive, then it is not invariably generated, nor is its commutator subgroup.

This prompts the following application of the main result from [ 41 ], and gives an example of behaviour of $G_n$ that makes it appear more similar to Thompson groups T and V than to F (see Subsection 7·6 below for another example of this).

7·6. Existence of free subgroups

As the result in the previous Subsection, the following is another example of a behaviour of dendrite rearrangement groups that resemble that of Thompson groups T and V rather than F.

This can be seen as a consequence of weak cell-transitivity (Proposition 7·11), since by [ 41 , Corollary 4·3] weakly cell-transitive rearrangement groups include non-abelian free groups. It can also be seen as a consequence of [ Reference Shi44 ], whose main result states that a group acting minimally on a non-degenerate dendrite includes non-abelian free groups. Minimality for dendrite rearrangement groups follows from [ 41 , Proposition 3·3] or can be proved using [ 28 , Theorem 5·2].

As a third option, here is a direct proof using the classical ping-pong argument. We need to find $g,h \in G_n$ and disjoint subsets X, Y of $D_n$ such that $g^z(X) \subseteq Y$ and $h^z(Y) \subseteq X$ , for all $z \in \mathbb{Z}\setminus\{0\}$ . Let $g \,:\!=\, g_0^2$ and let h be the element depicted in Figure 16 (the figure works for $n=3$ ; for higher n, one may consider any extension to the missing cells, which are those corresponding to words that include digits between 3 and $n-1$ ). Let $X = C^{\mathrm{o}}(1n) \cup C^{\mathrm{o}}(n)$ and $Y = C^{\mathrm{o}}(12) \cup C^{\mathrm{o}}(2)$ , which are disjoint. Now it is easy to check that $g(Y) \subseteq C^{\mathrm{o}}(1n)$ and that $g\left(C^{\mathrm{o}}(1n)\right) \subseteq C^{\mathrm{o}}(1n)$ , so $g^z(Y) \subseteq X$ for all $z\gt0$ . Analogous arguments show that $g^z(Y) \subseteq X$ for $z\lt0$ and that $h^z(X) \subseteq Y$ for all $z\gt0$ and $z\lt0$ .

Fig. 16. An element h that plays ping-pong with $g_0^2$ .

8·1. The Vicsek rearrangement groups

Introduced in [ Reference Belk and Forrest5 , Example 2·1], the Vicsek replacement systems are the same as the dendrite replacement systems $\mathcal{D}_n$ except for having an additional edge originating at the initial vertex. For example, Figure 17 depicts the Vicsek replacement system for $n=4$ .

Fig. 17. The Vicsek replacement system for $n=4$ .

It is clear that each Vicsek replacement system has the same Wa ${\dot {\mathrm z}}$ ewski dendrite $D_n$ as a limit space (although dendrite replacement systems $\mathcal{D}_n$ are arguably more natural replacement systems for these limit spaces, while the Vicsek replacement system reflects some of the metric aspects of the Vicsek fractals). It seems likely that the nth Vicsek rearrangement group is generated by a copy of the Higman–Thompson group $F_3$ along with the permutation group $\mathrm{Sym}(n)$ in a similar way in which $G_n$ is generated by F and $\mathrm{Sym}(n)$ by Theorem 4·3. It is known that F and $F_3$ are non-isomorphic groups, as they have different abelianisation, which was proved in [ Reference Brown13 ]. However, it does not seem immediately clear whether or not the nth Vicsek rearrangement group is isomorphic to $G_n$ , since Rubin’s theorem cannot be immediately applied (as discussed in Remark 7·5) and the finite subgroups (Subsection 7·1) are likely the same.

Additionally, one may be inspired by this example to build replacement systems with even more edges between the initial vertex $\iota$ and the branch point of the replacement graph, which may result in a rearrangement group that has some Higman–Thompson group $F_m$ acting on each EE-arc, in place of F or $F_3$ .

We did not investigate these questions any further.

8·2. Orientation-preserving dendrite rearrangement groups

The oriented dendrite replacement system $\boldsymbol{\mathcal{D}_n^+}$ , depicted in Figure 18, is the 2-coloured replacement system whose base graph and black replacement graph are the same graph consisting of a directed closed path of n red edges and n vertices, along with a black edge originating from each of these n vertices, each terminating at one of n other vertices. The initial and terminal vertices $\iota$ and $\tau$ of the black replacement graph are two distinct vertices of in-degree 1 and out-degree 0. The red replacement graph is a single red edge. If viewing without colors, simply keep in mind that edges labeled by r or r_r are red and the rest are black.

Fig. 18. A schematic depiction of the oriented dendrite replacement system $\mathcal{D}_n^+$ .

$\mathcal{D}_n^+$ is not an expanding replacement system (Definition 2·5), since the red replacement graph only has two vertices linked by an edge. Even if this does not pose an obstacle to the definition of a rearrangement group $G_n^+$ based on $\mathcal{D}_n^+$ (it would not be defined as a group of homeomorphisms, but only as a subgroup of the topological full group of a one-sided subshift of finite type), the gluing relation (Definition 2·6) might not be (and in fact is not) an equivalence relation, so the limit space cannot be defined in the usual way (Definition 2·7). However, since the gluing relation of $\mathcal{D}_n^+$ is reflexive and symmetric, it suffices to take the quotient of the symbol space of $\mathcal{D}_n^+$ by the transitive closure of the gluing relation in order to obtain an equivalence relation and a well defined limit space.

As noted in [ Reference Belk and Forrest5 , Remark 1·23], this limit space might not be as well-behaved as in the expanding case. In our case, if we say that an edge is null-expanding when the replacement graph of its colour consists of a sole edge, it is easy to see that each sequence $x \bar{r}$ , where xr is a null-expanding edge, ends up collapsing to a single point in the limit space, so the n red cells $C(y r_1), \dots, C(y r_n)$ (for any finite word y ending with a black edge) all degenerate to the same single point (these will correspond exactly to the branch points). It is easy then to check that the limit spaces of $\mathcal{D}_n$ and $\mathcal{D}_n^+$ are the same, so the possible issues of the limit space not being well-behaved do not show up here. More precisely, there is a canonical homeomorphism between the two limit spaces given by mapping each sequence $\omega$ of $\mathcal{D}_n^+$ that does not feature null-expanding edges to the same sequence of $\mathcal{D}_n$ . Every other sequences of $\mathcal{D}_n^+$ must be of the form $x r_i \bar{r}$ , so it represents the same branch point as $x i 1 \bar{n}$ and thus are mapped to this same sequence of $\mathcal{D}_n$ .

One can then define the rearrangement groups $G_n^+$ for the replacement systems $\mathcal{D}_n^+$ by associating to each graph pair diagram a homeomorphism in the usual way. The addition of the red edges will simply force every rearrangement to preserve the orientation of the dendrite, by which we mean that they preserve a circular ordering of the set of branches at each branch point. We leave the study of these orientation-preserving dendrite groups for future research, but we list here a few remarks about them:

1. (1) The group $G_3^+$ is likely isomorphic to the finitely generated Thompson-like group studied in the dissertation [ 46 ]. That is the group of those homeomorphisms of $S^1$ that preserve a lamination induced from the Julia set for the complex map $z \to z^2 + i$ (which is homeomorphic to $D_3$ ). We suspect that every $G_n^+$ can be built as a group of homeomorphisms of the circle by using laminations in this way.

2. (2) The edges of $\mathcal{D}_n^+$ are not undirected, unlike those of $\mathcal{D}_n$ (see Remark 3·1).

3. (3) The permutation subgroup from Subsubsection 3·3·1 translates to a subgroup of $G_n^+$ that is cyclic of order n instead of copies of the symmetric group on n elements.

4. (4) The Thompson subgroups from Lemma 4·1 do not seem to immediately translate to $G_n^+$ , but there are probably copies of Thompson’s group F inside $G_n^+$ that act on certain arcs of $D_n$ .

5. (5) The transitivity property described in Proposition 5·1 for $G_n$ may translate to an extension of isomorphisms between trees with a rotation system. If this happens, it might be used to prove that $G_n^+$ is dense in the group $\mathbb{H}_n^+$ of all orientation-preserving homeomorphisms of $D_n$ (i.e., the group of those homeomorphisms that at each branch point preserve a circular order of the branches).

6. (6) The parity map from Subsubsection 6·1·1 would be trivial in $G_n^+$ , so the abelianisation may simply be $\mathbb{Z}$ (which is what happens in the Airplane rearrangement group $T_A$ , see the next Subsection for the relation between $T_A$ and dendrites). Note that in [ 46 ] it is conjectured that the abelianisation of this dendrite-Thompson group is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_3$ , picturing a much more complex situation.

7. (7) If one chooses a different terminal vertex $\tau$ in the black replacement graph (Figure 18), then some of these remarks do not hold anymore (for example, if n is even and $\tau$ is the terminal vertex of the $n/2$ th edge, then the black edges are actually undirected).