von Neumann Algebras of Thompson-like Groups from Cloning Systems II

The author is indebted to Jon Bannon and Matthew Zaremsky for helpful discussions: to Jon Bannon for his help on the general theory of von Neumann algebras and recommending various avenues to explore; and to Matthew Zaremsky for his input on some of the technical aspects of $d$-ary cloning systems and helping tidy up some of the proofs.

Although this section is intended to be a relatively self-contained treatment of von Neumann algebras, for a excellent general reference overview we refer to Popa’s ICM survey (see [Pop07]), and for a more detailed introduction we refer to his book [AP18] with Anantharaman.

A von Neumann algebra is a $*$-subalgebra of bounded linear operators on some Hilbert space which is closed in the strong (equivalently, weak) operator topology. Arguably, the most well-studied and beloved von Neumann algebras arise from a group together with a unitary representation (e.g., the left-regular representation, the most natural unitary representation one can single out) or groups acting on measure spaces or other von Neumann algebras. When we have a group acting on a von Neumann algebra, we can form the so-called crossed product von Neumann algebra. Let us recall the basics of this construction. Throughout this paper, $G$ will almost exclusively denote a countable, discrete group. An action of $G$ on a von Neumann algebra $M$, which we will assume acts on a separable Hilbert space $\mathcal{H}$, is a homomorphism $\sigma:G\to\text{Aut}(M)$ of $G$ to the group $\text{Aut}(M)$ of (normal) $*$-automorphisms of $M$. Consider the Hilbert space $\ell^{2}(G,\mathcal{H})$ defined as follow:

equipped with the inner product

where the inner product $\langle,\rangle_{\mathcal{H}}$ on the right-hand side comes from the Hilbert space $\mathcal{H}$, and $\left\|~{}\right\|_{\mathcal{H}}$ denotes the norm on $\mathcal{H}$ induced by this inner product. Define $\lambda:G\to B(\ell^{2}(G,\mathcal{H}))$ and $\pi:M\to B(\ell^{2}(G,\mathcal{H}))$ by

for $g,h\in G$, $x\in M$, and $\psi\in\ell^{2}(G,\mathcal{H})$. One can easily verify that the former is a faithful unitary representation of $G$ while the latter is a faithful normal $*$-homomorphism, and that the commutation relation

must hold, where $x\in M$ and $g\in G$. Putting this together, the crossed product von Neumann algebra of $G$ acting on $M$, denoted by $M\rtimes_{\sigma}G$, is defined to be the von Neumann subalgebra of $B(\ell^{2}(G,\mathcal{H}))$ generated by $\pi(M)$ and $\pi(G)$; that is, form $M\rtimes_{\sigma}G$ by first forming the $*$-algebra generated by $\pi(M)$ and $\lambda(G)$ and then taking the weak or stong operator topology closure. We note that the $*$-isomorphism class of $M\rtimes_{\sigma}G$ is independent of the choice of separable Hilbert space $\mathcal{H}$ on which we represent $M$. At times we may omit the action $\sigma$ from the notation when the context is clear and simply write $M\rtimes G$.

If $M=\mathbb{C}$ with $G$ acting trivially, we obtain the so-called group von Neumann algebra of $G$, denoted as $L(G)$. If our group is given by a semi-direct product $N\rtimes_{\sigma}H$, then when we form the group von Neumann algebra of it, the semi-direct product translates into a crossed product von Neumann algebra. That is,

where $\widehat{\sigma}$ denotes the action of $H$ on $L(N)$ induced from the action $\sigma$ of $H$ on $N$.

Factor von Neumann algebras, those von Neumann algebras whose center consists only of scalar multiples of the identity, are the simple objects among von Neumann algebras and are therefore expectedly important in the theory. Indeed, all von Neumann algebras can be decomposed as a direct integral of factors. The theory reduces even further to so-called type $\operatorname{II}_{1}$ factors, meaning they essentially represent the final frontier in the classification of von Neumann algebras. Factors of type $\operatorname{II}_{1}$ are infinite-dimensional factors which admit a normal, finite, faithful, normalized trace $\tau$. Hence, it is helpful to know when a general procedure for constructing von Neumann algebras, such as the crossed product construction, yields a type $\operatorname{II}_{1}$ factor.

For crossed product von Neumann algebras, we have some nice standard criteria for when the crossed product is a type $\operatorname{II}_{1}$ factor in terms of the way by which the group acts. Before we can state these criteria, though, let us recall two important ways a group can act on a von Neumann algebra. If $\sigma$ is an action of group $G$ on a von Neumann algebra $M$, then the action $\sigma$ of $G$ on $M$ is said to be ergodic provided $M^{G}=\mathbb{C}1$, where

is the fixed-point subalgebra associated to the action. The action is said to be free provided the automorphism $\sigma_{g}$ is properly outer for every non-trivial element $g\in G$, which means that if $x\in M$ with $xy=\sigma_{g}(y)x$ for all $y\in M$, it follows that $x=0$. With these definitions in mind, we have the following standard criteria from [AP18]:

Given an inclusion of von Neumann algebras $N\subseteq M$, we can associate a certain group of unitaries in $M$ which stabilize the von Neumann subalgebra $N$ under the conjugation action:

Naturally, this is called the normalizer of $N$ in $M$, which contains $\mathcal{U}(N)$, the group of all unitary elements in $N$, as a normal subgroup. The normalizer of $N$ generates a von Neumann algebra, first by forming the $*$-algebra generated by the normalizer and then taking the weak (or strong) operator closure or the bicommutant, and the subalgebra $N$ can be classified in terms of the von Neumann algebra its normalizer generates. If the normalizer generates $N$, then $N$ is said to be singular; if it generates a proper subalgebra of $M$ other than $N$, then $N$ is said to be semi-regular; and, finally, if the normalizer generates $M$, then $N$ is said to be regular.

Now for some terminological conventions. If $H\leq G$ is an inclusion of groups, then we say that $H$ is self-normalizing in $G$ if the normalizer of $H$ in $G$ is “trivial;” i.e., if $\mathcal{N}_{G}(H)=H$. Similarly, if $N\subseteq M$ is an inclusion of von Neumann algebras, then we say that $N$ is self-normalizing $M$ if the normalizer of $N$ in $M$ is “trivial;” i.e., if $\mathcal{N}_{M}(N)=\mathcal{U}(N)$.

In addition to looking at normalizers, we can look at one-sided normalizers. The one-sided normalizer of $N$ in $M$ is the semigroup defined to be

which contains the usual von Neumann algebra normalizer. Given an inclusion $H\leq G$ of groups, we can similarly define the the normalizer and one-sided normalizer of $H$ in $G$ which as denoted as $\mathcal{N}_{G}(H)$ and $\mathcal{ON}_{G}(H)$, respectively.

Let us now turn to the notion of irreducibility.

Equivalently, $N\subseteq M$ is irreducible if and only if the conjugation action of the unitary group $\mathcal{U}(N)$ on $M$ is ergodic. For this reason, irreducible inclusions are also sometimes called ergodic inclusions or embeddings as in [Pop21]. With this definition of irreduciblity in mind, we can immediately rephrase Citation 1.1. First, the inclusion $M\subseteq M\rtimes_{\sigma}G$ is irreducible if and only if the action is free. Second, if $G$ is an ICC group, then the inclusion $L(G)\subseteq M\rtimes_{\sigma}G$ is irreducible if and only if the action of $G$ on $M$ is ergodic.

Irreducible inclusions of von Neumann algebras are nice for a variety of reasons, but for our purposes they are desirable for the following reasons. First, irreducible inclusions are nice because they allow us to “upgrade” factoriality of a subfactor to the containing or ambient von Neumann algebra. As a matter of fact, it follows quite easily from this definition that if $P$ is an intermediate von Neumann algebra of $N\subseteq M$, then the inclusion $P\subseteq M$ is also irreducible, and, moreover, if $N$ is a factor, then $P$ must also be a factor; in particular, $M$ must be a factor. Second, when the von Neumann algebras $N\subseteq M$ are group von Neumann algebras arising from an inclusion of groups, the von Neumann algebra normalizer admits a nice description in terms of the group normalizer, and determining whether the von Neumann subalgebra $N$ is regular, semi-regular, or singular is somewhat easier (see Citation 1.4).

It turns out we have some nice group-theoretic characterizations for when the group von Neumann algebra is a factor (and hence a type $\operatorname{II}_{1}$ factor) and when a group inclusion gives rise to an irreducible inclusion of von Neumann algebras. For factoriality of $L(G)$, it is a standard, classical fact that $L(G)$ is a factor if and only if $G$ is ICC, which means that the conjugacy class of any non-trivial element is infinite. Clearly, the property of being ICC is equivalent to the property that the only element with finite index centralizer is the identity. For irreducibility of an inclusion of group von Neumann algebras, we have the following from [SWW09] which in a sense generalizes the ICC condition:

Note that a non-trivial element $g\in G$ having infinitely many $H$-conjugates is equivalent to $g$ having infinite index centralizer in $H$, and when $H=G$ we obtain the usual conditions for when $L(G)$ is a type $\operatorname{II}_{1}$ factor.

From [SWW09], we also have a theorem about computing the von Neumann algebra normalizer when we have an irreducible inclusion arising from a group inclusion. This says that the von Neumann algebra normalizer can be directly computed from the group normalizer.

Note that if $\lambda:G\to\mathcal{U}(\ell^{2}(G))$ denotes the left-regular representation of $G$, defined on the canonical basis $\{\delta_{x}\}_{x\in G}$ of $\ell^{2}(G)$ as $\lambda(g)\delta_{x}:=\delta_{gx}$ for $g,x\in G$, then $\lambda_{g}:=\lambda(g)$ denotes the unitary element induced by $g\in G$. Roughly, this theorem states that the (one-sided) von Neumann algebra normalizers are made up of the (one-sided) group normalzers modulo a unitary from $L(H)$. Clearly the right-hand side is always contained in the left-hand side, but in the irreducible case we never have a strict inclusion but rather we always have equality. Note, for example, that if $H$ is self-normalizing in $G$, then the inclusion $L(H)\subseteq L(G)$ is singular.

The last property we discuss in this section is the weak asymptotic homomorphism property. First, given a von Neumann algebra $M$ with a normal, finite, faithful, normalized trace $\tau$, we can form the $2$-norm on $M$ defined by $\left\|x\right\|_{2}=\sqrt{\tau(x^{*}x)}$ for $x\in M$. For simplicity, in the future we will just refer to $\tau$ as a trace. Also, as an aside, we mention that given this norm $\left\|~{}\right\|_{2}$ on $M$, we can form the space $\ell^{\infty}(\mathbb{N},M)$ of all $\left\|~{}\right\|_{2}$-bounded sequences, which will be useful when stating the McDuff property and property Gamma in Section 1.4. With this in mind, the weak asymptotic homomorphism property is defined as follows.

As we noted in the introduction, proper finite index inclusions preclude the WAHP, meaning that if $N\subseteq M$ is a proper finite index inclusion, then $N\subseteq M$ cannot satisfy the WAHP (see [GW10]). The WAHP can also be rephrased in terms of the one-sided quasi-normalizer of $N$ in $M$ being “trivial,” and using this reformulation we can deduce two other easy consequences. The following comes from [FGS11].

An element $x\in M$ for which there exist $x_{1},\dots,x_{n}\in M$ with $Nx\subseteq\sum_{i=1}^{n}x_{i}N$ is called a one-sided quasi-normalizer of $N$, so the WAHP is equivalent saying that the one-sided quasi-normalizer of $N$ in $M$ is trivial. We let $\mathcal{QN}^{(1)}_{M}(N)$ denote the set of one-sided quasi-normalizers, and we have the following relationship between the various types of normalizers:

Using this formulation of the WAHP, we note that it is quite easy to see that if $N\subseteq M$ is an inclusion of von Neumann algebras satisfying the WAHP, then the inclusion must also be singular. More than that, the one-sided normalizer of $N$ in $M$ must be trivial (i.e., $\mathcal{ON}_{M}(N)=\mathcal{U}(N)$). Furthermore, it is quite easy to argue that if the inclusion $N\subseteq M$ satisfies the WAHP, then the inclusion must also be irreducible. We note that Theorem 3.1 in [FGS11] is actually phrased in terms of the relative weak asymptotic homomorphism property, but Citation 1.6 is easily derived from it.

As Jolissaint notes in [Jol12a], the inclusion $N\subseteq M$ satisfies the WAHP if and only if the inclusion $N\subseteq M$ is weakly mixing, i.e., $N$ is a weakly mixing subalgebra of $M$ (see [Jol12a, Theorem 1.4]). This equivalence is also true of the relative versions of the WAHP and weak mixing involving triples of von Neumann algebras, but we do not explore the relative versions in this paper. For more details on the relative WAHP and relative weak mixing, we refer the interested reader to [Jol12a], [Jol12b], [FGS11], and [CFM13].

In the case of an inclusion of von Neumann algebras arising from a group inclusion, condition (2) in Citation 1.6 translates to a condition about covering a right coset with left cosets. More precisely, we have the following from [FGS11]:

Similar to what we said above, an element $g\in G$ for which there are $g_{1},\dots,g_{n}\in G$ with $Hg\subseteq\bigcup_{i=1}^{n}g_{i}H$ is called a one-sided quasi-normalizer of $H$. Later in the paper, we may say that an inclusion $H\leq G$ of countable discrete groups satisfies the right coset covering condition or that the inclusion has the right coset covering property if condition (2) is satisfy, meaning that the inclusion $L(H)\subseteq L(G)$ satisfies the WAHP if and only if the inclusion $H\leq G$ satisfies the right coset covering condition or has the right coset covering property.

To summarize this subsection, we note that for an inclusion of tracial von Neumann algebras $N\subseteq M$, we have the $N\subseteq M$ satisfying the WAHP implies that it is singular and that it is irreducible. In the course of this paper, we will see many examples where all the implications are strict. For an inclusion which is irreducible but does not satisfy the WAHP (or is even singular), see the Röver–Nekrashevych groups in Section 3.2, and for an inclusion which is singular but not does not satisfy the WAHP, see Section 4.

Although we are not going to explore the three properties mentioned in the title of this section in relation to $d$-ary cloning systems in this paper as much as we did in [BZb], we do need them to state our results in that paper, particularly Citation 0.2, and then apply said results to prove that the Higman–Thompson groups $F_{d}$ are McDuff for all $d\geq 2$ in Section 5. Let us now define the (relative) McDuff property.

Recall that $R$ can be constructed by forming the group von Neumann algebra of $S_{\infty}$ (or any countable amenable ICC group for that matter), although it has other manifestations. It turns out that the McDuff property can be equivalently phrased in terms of central sequences.

Remarkably, the mere existence of a pair of non-trivial, inequivalent central sequences in a type $\operatorname{II}_{1}$ factor $M$ is equivalent to the existence of an isomorphism $M\to M\otimes R$, i.e., equivalent to $M$ being McDuff (see [McD70]). A related classical invariant of type $\operatorname{II}_{1}$ factors, which goes back to Murray and von Neumann, the founders of the theory of von Neumann algebras (see [MvN43]), is that of property Gamma.

Phrasing the McDuff property in terms of central sequences, it is clear that the McDuff property entails property Gamma. For an ICC group $G$, a classical result of Effros says that if $L(G)$ has property Gamma, then $G$ is inner amenable (see [Eff75]). Recall that a group $G$ is inner amenable if it admits a finitely additive, conjugation invariant probability measure on the set of subsets of $G\setminus\{e\}$. In summary, for a countable ICC group $G$ we have the following strict implications:

The first implication is easily seen to be strict: let $G$ be the direct product of any amenable ICC group with $\mathbb{F}_{2}$, the free group on two generators. As for the second implication being strict, the group von Neumann algebras of some of the Baumslag-Solitar groups have property Gamma (see [Sta06] and [hb]). However, Fima proved in [Fim11] that they also yield prime factors and hence cannot be McDuff factors. Recall that a von Neumann algebra $M$ is said to be prime if whenever $M$ can be decomposed into a tensor product of von Neumann algebras, one of the tensor product factors must be finite dimensional. Finally, Vaes constructed an example in [Vae12] showing that the third implication is strict, an implication which was long thought to be an equivalence.

In this section, we recall the basics of the $d$-ary cloning system construction which produces Thompson-like groups. We also recall some important properties of $d$-ary cloning systems and point out some examples. We now recall the definition of a $d$-ary cloning system and its axioms.

Given a $d$-ary cloning system $((G_{n})_{n\in\mathbb{N}},(\rho_{n})_{n\in\mathbb{N}},(\kappa_{k}^{n})_{k\leq n})$, we obtain a Thompson-like group, denoted as $\mathscr{T}_{d}(G_{*})$. When $d=2$, we simply write $\mathscr{T}(G_{*})$ for $\mathscr{T}_{2}(G_{*})$. Sometimes we may say that the maps $((\rho_{n})_{n\in\mathbb{N}},(\kappa_{k}^{n})_{k\leq n})$ “form” a $d$-ary cloning system on the sequence of groups $(G_{n})_{n\in\mathbb{N}}$ or that the sequence of groups $(G_{n})_{n\in\mathbb{N}}$ is “equipped” with the $d$-ary cloning system $((\rho_{n})_{n\in\mathbb{N}},(\kappa_{k}^{n})_{k\leq n})$ (as we already did in the introduction), which emphasizes the fact that a given sequence of groups may admit a variety of $d$-ary cloning systems, which we will see is certainly the case in Section 2.2.

Before we recall the construction of the Thompson-like group $\mathscr{T}_{d}(G_{*})$ arising from the $d$-ary cloning system $((G_{n})_{n\in\mathbb{N}},(\rho_{n})_{n\in\mathbb{N}},(\kappa_{k}^{n})_{k\leq n})$, a few remarks are in order concerning the $d$-ary cloning system axioms. Although the $d$-ary cloning maps are not homomorphisms, axiom (C1) says that they are twisted homomorphisms with the twisting given by the representation maps $\rho_{n}$ (i.e., the action of each $G_{n}$ on $\{1,\dots,n\}$). As for axiom (C2), this axiom tells us that the $d$-ary cloning maps satisfy certain commutation relations which essentially mirror the relations in the infinite presentation of the Higman–Thompson group $F_{d}$:

Finally, axiom (C3) essentially says that the $d$-ary cloning maps $(\kappa_{k}^{n})_{k\leq n}$ on $(G_{n})_{n\in\mathbb{N}}$ have to be compatible with the standard $d$-ary cloning maps $(\varsigma_{k}^{n})_{k\leq n}$ on the symmetric groups $(S_{n})_{n\in\mathbb{N}}$ in the sense that the two permutations $\rho_{n+d-1}((g)\kappa_{k}^{n})$ and $(\rho_{n}(g))\varsigma_{k}^{n}$ have to agree almost everywhere on $\{1,\dots,n\}$ for every $g\in G_{n}$ and every $n\in\mathbb{N}$. It turns out that in many important and canonical examples (e.g., Higman–Thompson groups and their braided variants) these two permutations will actually agree everywhere (see Section 2.2), but for the $d$-ary cloning system construction to produce a group with a well-defined binary operation it is not necessary that they agree everywhere.

Let us now recall the construction of the group $\mathscr{T}_{d}(G_{*})$. By a $d$-ary tree we mean a finite rooted tree in which each non-leaf vertex has $d$ children, and a $d$-ary caret is a $d$-ary tree with $d$ leaves. A leaf is any vertex with no outgoing edges, which have to exist since we are considering finite trees. For the sake of convenience, we may sometimes say “tree” instead of “$d$-ary tree” when there is no risk of confusion. If $T$ is any $d$-ary tree, we let $n(T)$ denote the number of leaves of $T$. Given a $d$-ary tree $T$, let us call an expansion of $T$ the result of gluing the root of a $d$-ary caret to one of its $n(T)$ leaves, and then recursively extend the definition of extension to mean the result adding $d$-ary carets to its leaves an arbitrary but finite number of times. Recursively extending the notion of expansion allows us to essentially “add” whole $d$-ary trees to any leaf of $T$ by gluing the root of the given $d$-ary tree to the desired leaf of $T$.

Let $((G_{n})_{n\in\mathbb{N}},(\rho_{n})_{n\in\mathbb{N}},(\kappa_{k}^{n})_{k\leq n})$ be a $d$-ary cloning system. Elements of $\mathscr{T}_{d}(G_{*})$ are equivalence classes represented by a triple $(T,g,U)$, where $T$ and $U$ are $d$-ary trees with the same number of leaves, say $n\in\mathbb{N}$, and $g$ is an element of $G_{n}$. Let us now explain the equivalence relation which gives rise to such equivalence classes (i.e., elements of $\mathscr{T}_{d}(G_{*})$). Given a triple $(T,g,U)$, an expansion of it is a triple of the form $(T_{\rho_{n}(g)(k)},(g)\kappa_{k}^{n},U_{k})$, where $U_{k}$ is obtained from $U$ by adding a $d$-ary caret to the $k$-th leaf and $T_{\rho_{n}(g)(k)}$ is obtained from $T$ by adding a $d$-ary caret to the $\rho_{n}(g)(k)$-th leaf. The equivalence relation on these triples is defined to be the symmetric and transitive closure of the expansions. Hence, roughly, two triples are equivalent if they differ by some finite sequence of expansions and reductions, where a reduction is simply a reverse expansion. We write $[T,g,U]$ for the equivalence class of $(T,g,U)$, which are the elements of $\mathscr{T}_{d}(G_{*})$.

We have described what the elements of $\mathscr{T}_{d}(G_{*})$ look like, but in order for this to form a group we must equip it with a binary operation. There is in fact a very natural binary operation on $\mathscr{T}_{d}(G_{*})$ and here is how to define it. Given any two elements $[T,g,U]$ and $[V,h,W]$ in $\mathscr{T}_{d}(G_{*})$, up to expansion we can assume $U=V$, because any pair (or even finite collection) of $d$-ary trees have a common expansion obtainable from either of them by adding the appropriate number of $d$-ary carets to the appropriate leaves (e.g., the their “union” is a common expansion). Now, multiplication of these two elements of $\mathscr{T}_{d}(G_{*})$ is given by cancelling out the common tree and multiplying the group elements; i.e.,,

The $d$-ary cloning axioms guarantee that this is a well-defined group operation. The identity is $[T,1,T]$ for any $d$-ary tree $T$, and group inversion is given by

As we alluded to in the introduction, we said that the group $\mathscr{T}_{d}(G_{*})$ can be regarded as a sort of Thompson-esque limit of the sequence of groups $(G_{n})_{n\in\mathbb{N}}$, and as such it should contain copies of the groups $G_{n}$ in the limit, as is the case with most limit constructions. Indeed, given any $d$-ary tree $T$, we can define

which is easily verified to be a subgroup of $\mathscr{T}_{d}(G_{*})$. Then $g\mapsto[T,g,T]$ defines a group embedding of $G_{n(T)}$ into $\mathscr{T}_{d}(G_{*})$ whose image is $G_{T}$.

If $((G_{n})_{n\in\mathbb{N}},(\rho_{n})_{n\in\mathbb{N}},(\kappa_{k}^{n})_{k\leq n})$ is any $d$-ary cloning system and $(H_{n})_{n\in\mathbb{N}}$ is a sequence of subgroups such that the $d$-ary cloning system is “compatible” with the sequence $(H_{n})_{n\in\mathbb{N}}$, in the sense that $(H_{n})\kappa_{k}^{n}\subseteq H_{n+d-1}$ for all $n\in\mathbb{N}$ and $1\leq k\leq n$, then the $d$-ary cloning system on $(G_{n})_{n\in\mathbb{N}}$ can be “restricted” to form a $d$-ary cloning system on $(H_{n})_{n\in\mathbb{N}}$. I.e., we obtain representation maps and $d$-ary cloning maps $\rho_{n}^{\prime}$ and $(\kappa_{k}^{n})^{\prime}$, respectively, on $(H_{n})_{n\in\mathbb{N}}$ by restricting $\rho_{n}$ and $\kappa_{k}^{n}$, respectively, and obtain a $d$-ary cloning system $((H_{n})_{n\in\mathbb{N}},(\rho_{n}^{\prime})_{n\in\mathbb{N}},((\kappa_{k}^{n})^{\prime})_{k\leq k})$ called a $d$-ary cloning subsystem of $((G_{n})_{n\in\mathbb{N}},(\rho_{n})_{n\in\mathbb{N}},(\kappa_{k}^{n})_{k\leq n})$. The resulting Thompson-like group $\mathscr{T}_{d}(H_{*})$ is a subgroup of $\mathscr{T}_{d}(G_{*})$, which is ensured by the fact that the $H_{n}$ are stable under applying the $d$-ary cloning maps.

Before we proceed any further, let us explain the standard $d$-ary cloning maps $\varsigma_{k}^{n}:S_{n}\to S_{n+d-1}$ on the sequence of finite symmetric groups referenced in the cloning axioms, and the Thompson-like group which results from them. Given $\sigma\in S_{n}$ a bijection of $\{1,\dots,n\}$, it can be represented as a diagram of arrows from one copy of $\{1,\dots,n\}$ up to a second copy with an arrow drawn from $i$ to $\sigma(i)$ for $i=1,\dots,n$. Then $(\sigma)\varsigma_{k}^{n}$, viewed in the same way as $\sigma$, is the diagram of arrows from $\{1,\dots,n,n+1,\dots,n+d-1\}$ up to a second copy of itself by taking the $k$-th arrow and replacing it by $d$ parallel arrows. See Figure 1 for an example, and for the rigorous, technical definition we refer to [SZ21, Example 2.2]. Although somewhat tedious, it is not difficult to verify that $((S_{n})_{n\in\mathbb{N}},(\text{id}_{S_{n}})_{n\in\mathbb{N}},(\varsigma_{k}^{n})_{k\leq n})$ forms a $d$-ary cloning system and that the resulting Thompson-like group is the Higman–Thompson group $V_{d}=\mathscr{T}_{d}(S_{*})$. Since the subgroups $\langle(1~{}2~{}\cdots~{}n)\rangle\leq S_{n}$ are stable under these $d$-ary cloning maps, we can restrict the standard $d$-ary cloning system on $(S_{n})_{n\in\mathbb{N}}$ to $(\langle(1~{}2~{}\cdots~{}n)\rangle)_{n\in\mathbb{N}}$ with the resulting Thompson-like group being the Higman–Thompson group $T_{d}=\mathscr{T}_{d}(\langle(1~{}2~{}\cdots~{}*)\rangle)$. Finally when restricted to the trivial subgroup, the resulting Thompson-like group is the Higman–Thompson group $F_{d}=\mathscr{T}_{d}(\{1\})$.