2.1 Braiding groups of automorphisms

Now we describe a braided version of all of the above. Let $B_d$ be the d-strand braid group. Via the standard projection $B_d\to S_d$ , we get an action of $B_d$ on X. Hence, we can consider finitely iterated wreath products $B_d\wr _X^n B_d$ , take the projective limit, and get the infinitely iterated wreath product

$$\begin{align*}B_d\wr_X (B_d \wr_X(B_d \wr_X\cdots)) \text{,} \end{align*}$$

which we may also write as $B_d \wr _X^\infty B_d$ .

Definition 2.4 (Braided $\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ )

Call the above infinitely iterated wreath product the braided automorphism group of $\mathcal {T}_d$ , denoted

Note that $\operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d) \cong B_d\wr _X \operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ , and so just like with $\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ , we get a “braided wreath recursion”: any $f\in \operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ decomposes as $f=\phi (f)(f_1,\dots ,f_d)$ , where

$$\begin{align*}\phi\colon \operatorname{br}\!\operatorname{\mathrm{Aut}}(\mathcal{T}_d)\to B_d \end{align*}$$

is the natural epimorphism coming from the wreath product, and $f_i\in \operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ . Also, note that the epimorphism $B_d\to S_d$ induces an epimorphism

$$\begin{align*}\pi\colon \operatorname{br}\!\operatorname{\mathrm{Aut}}(\mathcal{T}_d)\to \operatorname{\mathrm{Aut}}(\mathcal{T}_d)\text{.} \end{align*}$$

For any $f\in \operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ , define

where $f=\phi (f)(f_1,\dots ,f_d)$ is the braided wreath recursion. In particular, for any subgroup $G\le \operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ , the image $\psi (G)$ is a subset of $\operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)^X$ . This leads us to the following.

Note that the image in $\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ under $\pi $ of any braided self-similar group is a self-similar group (with an analogous statement for self-identical).

Before the next example, let us fix a generator $\zeta $ of $B_2\cong \mathbb {Z}$ , which will also be useful in all that follows. We will use the braid where the strand on the left crosses over the strand on the right as the strands go down, as in Figure 1.

Figure 1:

The braid $\zeta $ .

Before discussing our main example of the braided Grigorchuk group, let us pin down the kernel of the map $\pi \colon \operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)\to \operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ , i.e., $\pi \colon B_d\wr _X^\infty B_d\to S_d\wr _X^\infty S_d$ . First, note that the kernel of $\pi \colon B_d\to S_d$ is the pure braid group $PB_d$ . The action of $PB_d$ on X is trivial, so $PB_d \wr _X^n PB_d$ is simply a direct product of copies of $PB_d$ , namely $1+d+d^2+\cdots +d^n$ many copies. This equals the kernel of the natural map $B_d \wr _X^n B_d \to S_d \wr _X^n S_d$ . Now we get the following.

2.2 The braided Grigorchuk group

Now we construct a braided version of $\operatorname {\mathrm {Grig}}$ . Denote by a the element of $\operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_2)$ defined by the braided wreath recursion $a=\zeta (1,1)$ . Now define b, c, and d via the braided wreath recursions $b=(a,c)$ , $c=(a^{-1},d)$ , and $d=(1,b)$ .

Definition 2.7 (Braided Grigorchuk group)

The braided Grigorchuk group $\operatorname {\mathrm {brGrig}}$ is the subgroup of $\operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_2)$ defined by

The reason for using $c=(a^{-1},d)$ rather than $c=(a,d)$ is so that we get the pleasant-looking relation $bcd=1$ that holds analogously in the Grigorchuk group, as the proof of the next result shows.

Momentarily, we will prove that the braided Grigorchuk group is not finitely presented. The proof is inspired by the proof for the Grigorchuk group with some modifications, and we (roughly) follow this proof as given in [Reference DehornoydlH00]. First, we need some setup.

By Lemma 2.8, we see that there is a canonical epimorphism from onto $\operatorname {\mathrm {brGrig}}$ by mapping the generator of the first copy of $\mathbb {Z}$ to a and mapping the generators of $\mathbb {Z}^2$ to b and c. Thus any element in $\operatorname {\mathrm {brGrig}}$ can be written (non-uniquely) as

$$\begin{align*}z_1a^{k_1}z_2\cdots a^{k_\ell}z_{\ell+1}, \end{align*}$$

where the $z_i$ are of the form $b^{m_i}c^{n_i}$ for some $m_i, n_i \in \mathbb {Z}$ and are nontrivial except possibly when $i=1$ or $\ell +1$ . Call an expression of this form reduced. For a reduced expression as above, declare the length, denoted $|z_1a^{k_1}z_2\cdots a^{k_\ell }z_{\ell +1}|$ , to be the number of terms in the alternating product, so the length is $2\ell -1$ , $2\ell $ , or $2\ell +1$ depending on whether $z_1$ and/or $z_{\ell +1}$ are equal to $1$ . Similarly, define $|z_1a^{k_1}z_2\cdots a^{k_\ell }z_{\ell +1}|_a=\sum _{i=1}^\ell k_i$ , i.e., the sum of the exponents on the a terms.

Given any word w in the generators a, b, c, and d and their inverses, one can obtain a new word in reduced form by iteratively applying the following reductions and the analogous ones for their inverses:

1. (1) $g^ig^j=g^{i+j}$ for $g\in \{a,b, c, d\}.$

2. (2) $d^{-1}=bc.$

3. (3) $cb=bc.$

Call the resulting word $w^{\mathrm{red}}$ .

Lemma 2.11 Let w be a reduced expression for an element of $\operatorname {\mathrm {brGrig}}$ . Consider the braided wreath recursion $w=\phi (w)(w_1, w_2)$ . Then

$$\begin{align*}|w_i^{\mathrm{red}}|\leq \Big\lfloor \frac{|w|+1}{2} \Big\rfloor \text{. }\end{align*}$$

Proof Using the braided wreath recursion and applying the reductions, we get

$$\begin{align*}(b^kc^m)a^n= \begin{cases} \zeta^n(a^{k-m}, b^{-m}c^{k-m}), & \text{ if } n \text{ is even,} \\ \zeta^n(b^{-m}c^{k-m}, a^{k-m}), & \text{ if } n \text{ is odd.} \end{cases} \end{align*}$$

Thus, each pair $z_i a^{k_i}$ in the alternating product contributes at most one term to $w_1$ and at most one term to $w_2$ . The result now follows.

Let $W^{\mathrm{red}}$ be the set of reduced words, so we can identify $W^{\mathrm{red}}$ with $F=\mathbb {Z} \ast \mathbb {Z}^2$ . For $w\in W^{\mathrm{red}}$ and for $j_1, \dots , j_n\in \{1,2\}$ , let $w_{j_1\cdots j_n}^{\mathrm{red}}$ denote the reduced word defined inductively by

$$\begin{align*}w_{j_1\cdots j_n}^{\mathrm{red}}=((w_{j_1\cdots j_{n-1}})^{\mathrm{red}})_{j_n}^{\mathrm{red}} \text{.} \end{align*}$$

Let $\Psi\, {\colon}\, F\rightarrow \operatorname {\mathrm {brGrig}}$ be the canonical epimorphism.

Definition 2.8 For each $n\geq 0$ , let $K_n$ denote the subset of F given by

Proof It is clear that $K_n\leq K_{n+1}$ . The fact that $\bigcup \limits _{n=0}^\infty K_n=\ker {\Psi }$ follows from the fact that the procedure in Lemma 2.12 terminates. It remains to show that each $K_n$ is normal and that the inclusions are strict.

It is clear that $K_0$ is normal in F and so we proceed by induction. Observe that

$$\begin{align*}K_n=\{w\in F \mid |w|_a=0 \text{ and } w_1^{\mathrm{red}}, w_2^{\mathrm{red}}\in K_{n-1}\} \text{. }\end{align*}$$

Let $w\in K_n$ . We claim that any conjugate of w by one of the generators is again in $K_n.$ For $a^{-1}wa$ , this follows from the fact that $|a^{-1}wa|_a=|w|_a$ and that $(a^{-1}wa)_1=w_2$ and $(a^{-1}wa)_2=w_1$ . For the remaining cases, we see that $|b^{-1}wb|_a=|c^{-1}wc|_a=|d^{-1}wd|_a=|w|_a$ , and moreover

$$\begin{align*}(b^{-1}wb)_1=a^{-1}w_1 a, \qquad (b^{-1}wb)_2=c^{-1}w_2 c, \end{align*}$$

$$\begin{align*}(c^{-1}wc)_1=aw_1 a^{-1}, \qquad (c^{-1}wc)_2 =d^{-1}w_2 d, \end{align*}$$

$$\begin{align*}(d^{-1}wd)_1=w_1, \qquad (d^{-1}wd)_2=b^{-1}w_2 b \text{.} \end{align*}$$

Now normality follows from the induction hypothesis.

Finally, we verify that the inclusions are proper. Let $\sigma $ be the endomorphism of F defined by $\sigma (a)=a^{-1}c^{-1}a, \sigma (b)=d, \sigma (c)=b,$ and $\sigma (d)=c$ . Observe that $\sigma $ takes reduced words to reduced words, and moreover if $|w|_a=0$ then $|\sigma (w)|_a=0$ . Also, note that $\sigma (a)_1=d^{-1}$ , $\sigma (d)_1=a^{-1}$ , $\sigma (a)_2=a$ , and $\sigma (d)_2=d$ . Now fix

$$\begin{align*}w=[a,d][a^{-1},d^{-1}]=a^{-1}d^{-1}adada^{-1}d^{-1} \end{align*}$$

and

$$\begin{align*}\widetilde{w}=[d^{-1},a^{-1}][d,a]=dad^{-1}a^{-1}d^{-1}a^{-1}da \text{.} \end{align*}$$

We claim that $\sigma ^n(w)$ and $\sigma ^n(\widetilde {w})$ are in $K_{n+1}$ but not $K_n$ . For the base cases, one easily checks that w and $\widetilde {w}$ are both in $K_1$ but not in $K_0$ . Now note that $\sigma (w)_1=\widetilde {w}$ , $\sigma (w)_2=w$ , $\sigma (\widetilde {w})_1=w$ , and $\sigma (\widetilde {w})_2=\widetilde {w}$ , so using the base case, we get that $\sigma (w)$ and $\sigma (\widetilde {w})$ are both in $K_2$ but not $K_1$ . Continuing in this way, the result follows by induction.

Proof Suppose $\operatorname {\mathrm {brGrig}}$ is finitely presented. As it is a quotient of F, it has a presentation of the form

$$\begin{align*}\langle a,b,c,d \mid bcd,b^{-1}c^{-1}bc, r_1, r_2, \dots r_k\rangle\text{,} \end{align*}$$

giving $\operatorname {\mathrm {brGrig}}\cong F/R$ where $R=\ker (\Psi )$ is the normal closure of $r_1, \dots , r_k$ in F. As each $r_i$ is contained in some $K_n$ , this contradicts Lemma 2.13.

Anthony Genevois has pointed out to us that Proposition 2.14 also follows from [Reference Bestvina and BradyBGdlH13, Theorem 1.6], which implies that no finitely presented amenable group has $\operatorname {\mathrm {Grig}}$ as a quotient.

3.1 Cloning systems

Definition 3.1 (d-ary cloning system)

Let $d\ge 2$ be an integer, and let $(G_n)_{n\in \mathbb {N}}$ be a family of groups. For each $n\in \mathbb {N}$ , let $\rho _n\, {\colon}\, G_n\to S_n$ be a homomorphism to the symmetric group $S_n$ , called a representation map. For each $1\le k\le n$ , let $\kappa _k^n\, {\colon}\, G_n \to G_{n+d-1}$ be an injective function (not necessarily a homomorphism), called a d-ary cloning map. We write $\rho _n$ to the left of its input and $\kappa _k^n$ to the right of its input, for reasons of visual clarity. Now we call the triple

$$\begin{align*}((G_n)_{n\in\mathbb{N}},(\rho_n)_{n\in\mathbb{N}},(\kappa_k^n)_{k\le n}) \end{align*}$$

a d-ary cloning system if the following axioms hold: (C1): (Cloning a product) $(gh)\kappa _k^n = (g)\kappa _{\rho _n(h)k}^{n} (h)\kappa _k^n$ . (C2): (Product of clonings) $\kappa _\ell ^n \circ \kappa _k^{n+d-1} = \kappa _k^n \circ \kappa _{\ell +d-1}^{n+d-1}$ . (C3): (Compatibility) $\rho _{n+d-1}((g)\kappa _k^n)(i) = (\rho _n(g))\varsigma _k^n(i)$ for all $i\ne k,k+1,\dots ,k+d-1$ .

Here, we always have $1\le k<\ell \le n$ and $g,h\in G_n$ , and $\varsigma _k^n$ denotes the standard d-ary cloning maps for the symmetric groups.

The maps $\varsigma _k^n$ are explained in [Reference SteinSZ21, Example 2.2], and we will review them in Example 3.2.

Remark 3.1 For an illustration of a cloning map in the special case $G_n=\operatorname {br}\!W_n$ (defined in Section 3.3), we direct the reader to Figure 2.

Figure 2:

An example of $2$ -ary cloning on $\operatorname {br}\!W_2$ . Here, $f\in \operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_2)$ satisfies the braided wreath recursion $f=\zeta (f,f)$ . The picture shows that $(\zeta (\operatorname {\mathrm {id}},f))\kappa _2^2 = (\zeta )\vartheta _2^2\phi ^{(2)}(f)(\operatorname {\mathrm {id}},f,f)$ . We use thick lines to indicate the strand getting cloned and the resulting strands.

Given a d-ary cloning system on a family of groups $(G_n)_{n\in \mathbb {N}}$ , one gets a Thompson-like group, denoted $\mathscr {T}_d(G_*)$ , which can be viewed as a sort of “Thompson limit” of the $G_n$ . Let us recall the construction of $\mathscr {T}_d(G_*)$ . First, a d-ary tree is a finite rooted tree in which each nonleaf vertex has d children, and a d-ary caret is a d-ary tree with d leaves. An element of $\mathscr {T}_d(G_*)$ is represented by a triple $(T_-,g,T_+)$ where $T_\pm $ are d-ary trees with the same number of leaves, say n, and g is an element of $G_n$ . There is an equivalence relation on such triples, and the equivalence classes are the elements of $\mathscr {T}_d(G_*)$ . The equivalence relation is given by expansion and reduction: an expansion of $(T_-,g,T_+)$ is a triple of the form $(T_-',(g)\kappa _k^n,T_+')$ where $T_+'$ is $T_+$ with a d-ary caret added to the kth leaf and $T_-'$ is $T_-$ with a d-ary caret added to the $\rho _n(g)(k)$ th leaf. A reduction is the reverse of an expansion. Now declare that two triples are equivalent if we can get from one to the other via a finite sequence of expansions and reductions, and write $[T_-,g,T_+]$ for the equivalence class of $(T_-,g,T_+)$ .

We have not explained the group operation on $\mathscr {T}_d(G_*)$ . The idea is that given any two elements $[T_-,g,T_+]$ and $[U_-,h,U_+]$ , up to expansions, we can assume that ${T_+=U_-}$ . This is because any pair of d-ary trees have a common d-ary tree obtainable from either of them by adding d-ary carets to their leaves. Now the group operation on $\mathscr {T}_d(G_*)$ is defined by

when $T_+=U_-$ . The cloning axioms ensure that this is a well-defined group operation. The identity is $[T,1,T]$ (for any T) and inverses are given by $[T_-,g,T_+]^{-1} = [T_+,g^{-1},T_-]$ .

Example 3.2 (Cloning permutations)

The most fundamental example of a d-ary cloning system is on the family $(S_n)_{n\in \mathbb {N}}$ of symmetric groups. Take $\rho _n \, {\colon}\, S_n\to S_n$ to be the identity, and let

$$\begin{align*}\varsigma_k^n \, {\colon}\, S_n \to S_{n+d-1} \end{align*}$$

be the function described as follows. Visualize $\sigma \in S_n$ by drawing arrows going up, from a line of labels $1$ to n for the domain to another line of labels $1$ to n for the range, with an arrow from i to $\sigma (i)$ for each i. Now $(\sigma )\varsigma _k^n \in S_{n+d-1}$ is obtained by replacing the arrow from k to $\sigma (k)$ by d parallel arrows, and relabeling everything appropriately. For a (complicated) rigorous formula, see [Reference SteinSZ21, Example 2.2]. It turns out that this defines a d-ary cloning system, with d-ary cloning maps $\varsigma _k^n$ . The Thompson-like group $\mathscr {T}_d(S_*)$ that arises from this d-ary cloning system is (isomorphic to) the Higman–Thompson group $V_d$ .

Example 3.3 (Cloning braids)

The braided version of the above example is a d-ary cloning system on the family $(B_n)_{n\in \mathbb {N}}$ of braid groups. Take $\rho _n\, {\colon}\, B_n\to S_n$ to be the natural projection of $B_n$ onto $S_n$ , and let

$$\begin{align*}\vartheta_k^n \, {\colon}\, B_n \to B_{n+d-1} \end{align*}$$

be the function described as follows. Visualize $\beta \in B_n$ as an n-strand braid diagram, counting the strands $1$ to n at the bottom. Now $(\beta )\vartheta _k^n \in B_{n+d-1}$ is obtained by replacing the kth strand (counting at the bottom) by d parallel strands. As discussed in [Reference Witzel and ZaremskyWZ18, Remark 2.10], in the $d=2$ case, this really defines a cloning system (this was essentially already shown by Brin in [Reference BrinBri07], before cloning systems had been introduced, using the language of Zappa–Szép products), and it is easy to see that in the arbitrary d case, it defines a d-ary cloning system. The Thompson-like group $\mathscr {T}_d(B_*)$ that arises from this d-ary cloning system is (isomorphic to) the braided Higman–Thompson group $\operatorname {br}\!V_d$ , considered previously by Aroca and Cumplido in [Reference AllcockAC22], and the first author and Wu in [Reference Skipper and ZaremskySW].

3.2 Almost-automorphisms

Every automorphism of $\mathcal {T}_d$ induces a self-homeomorphism of the boundary $\partial \mathcal {T}_d$ , which is a d-ary Cantor space. The idea behind almost-automorphisms of $\mathcal {T}_d$ is to consider self-homeomorphisms of $\partial \mathcal {T}_d$ that locally “look like” they came from $\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ . Far more detail and rigor can be found, for example, in [Reference LeBoudecLB17] and [Reference SteinSZ21, Section 1.3]. For our purposes here, we will use the isomorphism in [Reference SteinSZ21, Theorem 2.6] to simply define the group of almost-automorphisms of $\mathcal {T}_d$ as the Thompson-like group arising from the following cloning system.

For $n\in \mathbb {N}$ , let

(in [Reference SteinSZ21], this was denoted $A_n$ , but here we will use $W_n$ ). Here, the wreath product has no subscript, and so this should be interpreted as meaning there are n copies of $\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ and $S_n$ acts on $\{1,\dots ,n\}$ in the standard way. Following [Reference SteinSZ21, Section 2.2], we will define a d-ary cloning system on $(W_n)_{n\in \mathbb {N}}$ . Let $\rho _n \, {\colon}\, W_n \to S_n$ be the natural epimorphism onto the $S_n$ term, i.e.,

To define the d-ary cloning maps $\kappa _k^n \, {\colon}\, W_n \to W_{n+d-1}$ , we need some notation. Given $\sigma (f_1,\dots ,f_n)\in W_n$ and $1\le k\le n$ , write the wreath recursion of $f_k$ as ${f_k=\rho (f_k)(f_k^1,\dots ,f_k^d)}$ . Also, write $\rho ^{(k)}(f_k)\in S_{n+d-1}$ for the image of $\rho (f_k)\in S_d$ under the (k-dependent) monomorphism $S_d\to S_{n+d-1}$ induced by the inclusion $\{1,\dots ,d\}\to \{1,\dots ,n\}$ sending j to $k+j-1$ . Now we can define $\kappa _k^n$ as follows:

As proved in [Reference SteinSZ21, Proposition 2.4], these $\rho _n$ and $\kappa _k^n$ define a d-ary cloning system on the $W_n$ .

Definition 3.3 (Group of almost-automorphisms)

The group of almost-automorphisms $\operatorname {\mathrm {AAut}}(\mathcal {T}_d)$ of $\mathcal {T}_d$ is

where $(W_n)_{n\in \mathbb {N}}$ is equipped with the above d-ary cloning system.

Now, for self-similar $G\le \operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ , consider the family $(S_n \wr G)_{n\in \mathbb {N}}$ . Thanks to self-similarity, the restriction of $\kappa _k^n$ to $S_n \wr G$ lands in $S_{n+d-1} \wr G$ , which means that the d-ary cloning system on the $W_n=S_n \wr \operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ restricts to a d-ary cloning system on $S_n \wr G$ .

Definition 3.4 (Röver–Nekrashevych group)

For a self-similar group $G\le \operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ , the Röver–Nekrashevych group for G is

where $(S_n\wr G)_{n\in \mathbb {N}}$ is equipped with the above d-ary cloning system.

This definition agrees with the usual definition first given by Nekrashevych [Reference NekrashevychNek04], thanks to [Reference SteinSZ21, Corollary 2.7].

The Röver group $V_2(\operatorname {\mathrm {Grig}})$ was first constructed by Röver in [Reference RöverRöv99], and generalized by Nekrashevych in [Reference NekrashevychNek04] to the full family of Röver–Nekrashevych groups $V_d(G)$ . Röver proved that $V_2(\operatorname {\mathrm {Grig}})$ is isomorphic to the abstract commensurator of $\operatorname {\mathrm {Grig}}$ , and is a finitely presented simple group [Reference RöverRöv02, Reference RöverRöv99]. Belk and Matucci [Reference Brady, Burillo, Cleary and SteinBM16] proved that it is even of type $\operatorname {\mathrm {F}}_\infty $ . Analogous results for certain $V_d(G)$ were proved by Nekrashevych [Reference NekrashevychNek04], Farley and Hughes [Reference Farley and HughesFH15], and the authors [Reference SteinSZ21].

3.3 Braiding groups of almost-automorphisms

Now we will introduce braided Röver–Nekrashevych groups. Starting with self-identical groups, this was previously done by Aroca and Cumplido in [Reference AllcockAC22], but for self-similar groups that are not self-identical, to the best of our knowledge, this has not been done. In particular, using the braided Grigorchuk group to construct a braided Röver group is new. As a remark, Aroca and Cumplido’s constructions could have been phrased in the language of cloning systems, as they mention in their introduction (though they did not use this framework outside their introduction), so one can view our approach here as a direct generalization of theirs.

Convention: The notation $B_n \wr G$ , i.e., with no subscript on the $\wr $ , will always mean $B_n \wr _{\{1,\dots ,n\}} G$ defined via the action of the braid group $B_n$ on $\{1,\dots ,n\}$ coming from the natural projection $B_n \to S_n$ .

Recall that $\operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)=B_d \wr _X^\infty B_d$ , and let

. We want to define a d-ary cloning system on $(\operatorname {br}\!W_n)_{n\in \mathbb {N}}$ . We will use the notation $\rho _n$ and $\kappa _k^n$ like we did in the nonbraided case for the cloning system on $(W_n)_{n\in \mathbb {N}}$ , and no confusion should arise. Let $\rho _n\, {\colon}\, \operatorname{br}\!W_n \to S_n$ be the composition of $\operatorname {br}\!W_n \to W_n$ with $W_n \to S_n$ , where the first map is induced by the standard epimorphism $B_n\to S_n$ together with $\pi \, {\colon}\, \operatorname{br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)\to \operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ , and the second map is the $\rho _n$ epimorphism from the nonbraided case. To define the d-ary cloning maps $\kappa _k^n \, {\colon}\, \operatorname{br}\!W_n \to \operatorname {br}\!W_{n+d-1}$ , we need some notation. Given $\beta (f_1,\dots ,f_n)\in \operatorname {br}\!W_n$ and $1\le k\le n$ , write the braided wreath recursion of $f_k$ as $f_k=\phi (f_k)(f_k^1,\dots ,f_k^d)$ . Also, write $\phi ^{(k)}(f_k)\in B_{n+d-1}$ for the image of $\phi (f_k)\in B_d$ under the (k-dependent) monomorphism $B_d\to B_{n+d-1}$ induced by adding $k-1$ new unbraided strands on the left and $n-k$ new unbraided strands on the right. Now we can define $\kappa _k^n$ as follows:

Note that $\kappa _k^n$ is injective, since $\vartheta _k^n$ is injective and $f_k$ is uniquely determined by its braided wreath recursion. See Figure 2 for example.

Proof The proof is similar to that of [Reference SteinSZ21, Proposition 2.4] about the nonbraided situation, and some parts work in exactly the same way. Hence, in the course of this proof, we will sometimes just state that a certain step works analogously to the corresponding step in [Reference SteinSZ21, Proposition 2.4]. First, we prove (C1) (cloning a product). Let $f=\beta (f_1,\dots ,f_n)$ and $g=\gamma (g_1,\dots ,g_n)$ be elements of $\operatorname {br}\!W_n$ , so the product $fg$ equals $\beta \gamma (f_{\gamma (1)}g_1,\dots ,f_{\gamma (n)}g_n)$ . Here, $B_n$ acts on $\{1,\dots ,n\}$ via the projection $B_n\to S_n$ , and the notation $\gamma (i)$ indicates this action. Similar to the proof of [Reference SteinSZ21, Proposition 2.4], the left-hand side of (C1) is

$$ \begin{align*} &(fg)\kappa_k^n = (\beta\gamma)\vartheta_k^n \phi^{(k)}(f_{\gamma(k)}g_k)\\ &\quad\times (f_{\gamma(1)}g_1,\dots,f_{\gamma(k-1)}g_{k-1},f_{\gamma(k)}^{\phi(g_k)(1)}g_k^1,\dots,f_{\gamma(k)}^{\phi(g_k)(d)}g_k^d,f_{\gamma(k+1)}g_{k+1},\dots,f_{\gamma(n)}g_n) \text{.} \end{align*} $$

Here, the braided wreath recursions of $f_{\gamma (k)}$ and $g_k$ are $f_{\gamma (k)}=\phi (f_{\gamma (k)})(f_{\gamma (k)}^1,\dots ,f_{\gamma (k)}^d)$ and $g_k=\phi (g_k)(g_k^1,\dots ,g_k^d)$ , so the braided wreath recursion of $f_{\gamma (k)}g_k$ is

$$\begin{align*}f_{\gamma(k)}g_k = \phi(f_{\gamma(k)}g_k)(f_{\gamma(k)}^{\phi(g_k)(1)}g_k^1,\dots,f_{\gamma(k)}^{\phi(g_k)(d)}g_k^d) \text{.} \end{align*}$$

Now we need to show that the right-hand side of (C1), which is $(f)\kappa _{\gamma (k)}^n(g)\kappa _k^n$ , equals the same thing. One can compute (similar to the proof of [Reference SteinSZ21, Proposition 2.4]) that this equals

$$ \begin{align*} &(\beta)\vartheta_{\gamma(k)}^n \phi^{(\gamma(k))}(f_{\gamma(k)}) (\gamma)\vartheta_k^n \phi^{(k)}(g_k) \\ &\times (f_1,\dots,f_{\gamma(k)-1},f_{\gamma(k)}^1,\dots,f_{\gamma(k)}^d,f_{\gamma(k)+1},\dots,f_n)^{(\gamma)\vartheta_k^n \phi^{(k)}(g_k)} \\ &\times (g_1,\dots,g_{k-1},g_k^1,\dots,g_k^d,g_{k+1},\dots,g_n) \text{,} \end{align*} $$

where the superscript indicates conjugation in $\operatorname {br}\!W_{n+d-1}$ . By the same argument as in the proof of [Reference SteinSZ21, Proposition 2.4], the “tuple parts” of the left- and right-hand sides of (C1) are the same, so we only need to show that the “braid parts” are the same, i.e., that $(\beta \gamma )\vartheta _k^n \phi ^{(k)}(f_{\gamma (k)}g_k) = (\beta )\vartheta _{\gamma (k)}^n \phi ^{(\gamma (k))}(f_{\gamma (k)}) (\gamma )\vartheta _k^n \phi ^{(k)}(g_k)$ . Since we already know the $\vartheta _k^n$ define a d-ary cloning system on braid groups, we know $(\beta \gamma )\vartheta _k^n=(\beta )\vartheta _{\gamma (k)}^n(\gamma )\vartheta _k^n$ , and since $\phi $ is a homomorphism, this means that it suffices to show that $(\gamma )\vartheta _k^n \phi ^{(k)}(f_{\gamma (k)}) = \phi ^{(\gamma (k))}(f_{\gamma (k)}) (\gamma )\vartheta _k^n$ . To see this, note that the kth to $(k+d-1)$ st strands of $(\gamma )\vartheta _k^n$ are all parallel to each other, and these are the only strands of $\phi ^{(k)}(f_{\gamma (k)})$ that can braid nontrivially (see Figure 3).

Figure 3:

An example of the last step of the verification of (C1) in the proof of Proposition 3.4. Here, $d=3$ , $k=3$ , and $n=5$ .

Next, (C2) follows by an exactly analogous argument to the proof of [Reference SteinSZ21, Proposition 2.4].

Finally, we turn to (C3). Let $\beta (f_1,\dots ,f_n)\in \operatorname {br}\!W_n$ and $i\ne k,k+1,\dots ,k+d-1$ . We need to show that $\rho _{n+d-1}((\beta (f_1,\dots ,f_n))\kappa _k^n)(i) = (\rho _n(\beta (f_1,\dots ,f_n)))\varsigma _k^n(i)$ . Setting $\sigma =\pi (\beta )$ , the right-hand side just equals $(\sigma )\varsigma _k^n(i)$ . The left-hand side equals

$$\begin{align*}\rho_{n+d-1}((\beta)\vartheta_k^n\phi^{(k)}(f_k)(f_1,\dots,f_{k-1},f_k^1,\dots,f_k^d,f_{k+1},\dots,f_n))(i) \text{,} \end{align*}$$

which is $\pi ((\beta )\vartheta _k^n\phi ^{(k)}(f_k))(i)$ . Since $\pi (\phi ^{(k)}(f_k))(i)=i$ (by virtue of $i\ne k,k+1,\dots ,k+d-1$ ), this equals $\pi ((\beta )\vartheta _k^n)(i)$ . Since $\pi $ plays the role of $\rho _n$ in the d-ary cloning system on the braid groups using the d-ary cloning maps $\vartheta _k^n$ , by (C3) for that d-ary cloning system, we know $\pi ((\beta )\vartheta _k^n)(i) = (\sigma )\varsigma _k^n(i)$ as desired.

The Thompson-like group $\mathscr {T}_d(\operatorname {br}\!W_*)$ is a braided version of $\mathscr {T}_d(W_*)=\operatorname {\mathrm {AAut}}(\mathcal {T}_d)$ , and we will denote it by $\operatorname {br}\!\operatorname {\mathrm {AAut}}(\mathcal {T}_d)$ .

Now, for braided self-similar $G\le \operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ , consider the family $(B_n \wr G)_{n\in \mathbb {N}}$ . Thanks to braided self-similarity, the restriction of $\kappa _k^n$ to $B_n \wr G$ lands in $B_{n+d-1} \wr G$ , which means that the d-ary cloning system on the $\operatorname {br}\!W_n=B_n \wr \operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ restricts to a d-ary cloning system on the $B_n \wr G$ .

Definition 3.6 (Braided Röver–Nekrashevych group)

For a braided self-similar group $G\le \operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ , the braided Röver–Nekrashevych group for G is

where $(B_n\wr G)_{n\in \mathbb {N}}$ is equipped with the above d-ary cloning system.

Our main example is the following.

See Figure 4 for an example of an element of the braided Röver group.

Figure 4:

An element of the braided Röver group. Here, we draw a triple $(T_-,\beta (f_1,\dots ,f_m),T_+)$ with $T_+$ upside down so that $\beta $ is a braid from the leaves of $T_-$ to the leaves of $T_+$ , and the $f_i$ label the strands. The element equals $[\wedge ,(a,b),\wedge ]$ , for $\wedge $ the tree with one caret. As indicated in the picture, after expansions, this is the same element as $[T,\beta (1,1,a,c),T]$ for T the result of adding a caret to each leaf of $\wedge $ and $\beta \in B_4$ the braid crossing the first strand over the second.

There is an obvious relationship between braided Röver–Nekrashevych groups and (nonbraided) Röver–Nekrashevych groups, analogous to how the braided Thompson group $\operatorname {br}\!V$ surjects onto Thompson’s group V. Given a braided Röver–Nekrashevych group $\operatorname {br}\!V_d(G)$ , we get a well-defined epimorphism

$$\begin{align*}\pi\, {\colon}\, \operatorname{br}\!V_d(G)\to V_d(\pi(G)) \text{,} \end{align*}$$

where $\pi (G)$ is the image of G under $\pi \, {\colon}\, \operatorname{br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d) \to \operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ . This epimorphism (which by abuse of notation we are also denoting by $\pi $ ) is given by sending $[T_-,\beta (f_1,\dots ,f_n),T_+]$ to $[T_-,\pi (\beta )(\pi (f_1),\dots ,\pi (f_n)),T_+]$ . The d-ary cloning systems on $(B_n\wr G)_{n\in \mathbb {N}}$ and $(S_n\wr \pi (G))_{n\in \mathbb {N}}$ are clearly respected by $\pi $ , so this map really is well defined.

4.1 Stein–Farley complexes

The starting point for deducing finiteness properties for a Thompson-like group is often to construct a so-called Stein–Farley complex on which the group can act. For groups of the form $\mathscr {T}_2(G_*)$ , i.e., those arising from ( $2$ -ary) cloning systems, this was done in [Reference Witzel and ZaremskyWZ18]. For groups of the form $\mathscr {T}_d(S_*\wr G)$ , i.e., Röver–Nekrashevych groups, the Stein–Farley construction was done in [Reference SteinSZ21]. For arbitrary groups of the form $\mathscr {T}_d(G_*)$ , i.e., those arising from d-ary cloning systems in general, the Stein–Farley construction has not technically been done in the literature, but it is easy to mimic the $2$ -ary construction, and this is the goal of this subsection. Also, see [Reference ZaremskyZar21] for an introductory take on the situation when $d=2$ and $G_m=\{1\}$ for all m, i.e., for Thompson’s group F. The name Stein–Farley complex pays homage to Stein [Reference Skipper and WuSte92] and Farley [Reference FarleyFar03], who constructed complexes like these for the classical Thompson groups and some close relatives.

Given a d-ary cloning system $((G_m)_{m\in \mathbb {N}},(\rho _m)_{m\in \mathbb {N}},(\kappa _k^m)_{k\le m})$ , we will construct a cube complex, denoted by $\mathscr {X}_d(G_*)$ , called the Stein–Farley complex for the d-ary cloning system. This is done in a number of steps.

A groupoid: First, we consider the set of equivalence classes $[F_-,g,F_+]$ , where $F_-$ is a d-ary forest, say with m leaves (and some number of roots), g is an element of $G_m$ , and $F_+$ is a d-ary forest with m leaves (and some number of roots). The equivalence relation on such triples is given by the expansion and reduction moves, as with elements of $\mathscr {T}_d(G_*)$ . This set is a groupoid; two elements $[F_-,g,F_+]$ and $[E_-,h,E_+]$ can be multiplied whenever the number of roots of $F_+$ equals the number of roots of $E_-$ . In this case, up to expansions, we can assume that $F_+=E_-$ , and the multiplication in this groupoid works analogously to the multiplication in $\mathscr {T}_d(G_*)$ . Multiplication is well defined in the groupoid for all the same reasons that it is well defined in $\mathscr {T}_d(G_*)$ . Denote this groupoid by $\mathscr {G}_d(G_*)$ .

A poset: Now we restrict to equivalence classes of triples of the form $[T_-,g,F_+]$ for $T_-$ a d-ary tree, and mod out an additional equivalence relation given by right multiplication by elements of the form $[1,g',1]$ , for $1$ a trivial forest with some number of roots, m, and $g'\in G_m$ . Denote the resulting equivalence classes by $[T_-,g,F_+]_G$ , and write $\mathscr {P}_d(G_*)$ for the set of all of them. Given $[T_-,g,F_+]_G\in \mathscr {P}_d(G_*)$ , say with $F_+$ having m roots, and a groupoid element of the form $[F,1,1]$ for F a d-ary forest with m roots, it makes sense to consider the product $[T_-,g,F_+][F,1,1]$ and the equivalence class $[T_-,g,F_+][F,1,1]_G$ . We define a partial order $\le $ on $\mathscr {P}_d(G_*)$ by declaring

$$\begin{align*}[T_-,g,F_+]_G\le [T_-,g,F_+][F,1,1]_G \text{.} \end{align*}$$

(Note that thanks to the equivalence relation, in fact, we have

$$\begin{align*}[T_-,g,F_+]_G\le [T_-,g,F_+][1,g',1][F,1,1]_G \end{align*}$$

for any relevant $g'$ .) It is easy to see that $\le $ is reflexive, transitive, and antisymmetric, and hence really is a partial order. It is also clear that $\mathscr {P}_d(G_*)$ with $\le $ is a directed poset, since for any $[T_-,g,F_+]_G$ we have $[T_-,g,F_+]_G \le [T_-,g,1]_G=[T_-,1,1]_G$ , and any two $[T_-,1,1]_G$ and $[T_-',1,1]_G$ have an upper bound. Hence, the geometric realization $|\mathscr {P}_d(G_*)|$ is contractible.

A cube complex: Now we construct the Stein–Farley complex $\mathscr {X}_d(G_*)$ , which is a cube complex having a natural subdivision identifiable with a certain subcomplex of $|\mathscr {P}_d(G_*)|$ . The vertex set of $\mathscr {X}_d(G_*)$ is the whole vertex set of $|\mathscr {P}_d(G_*)|$ , namely $\mathscr {P}_d(G_*)$ . Let E be a d-ary forest whose trees each have at most one d-caret; call such a d-ary forest elementary. If E is an elementary d-ary forest with one nontrivial tree and with the same number of roots as $F_+$ , we can consider the product $[T_-,g,F_+][E,1,1]$ in the groupoid $\mathscr {G}_d(G_*)$ . We declare that the vertices $[T_-,g,F_+]_G$ and $[T_-,g,F_+][E,1,1]_G$ span an edge in $\mathscr {X}_d(G_*)$ . If $[T_-,g,F_+]_G\le [T_-,g,F_+][E,1,1]_G$ for E elementary, write

$$\begin{align*}[T_-,g,F_+]_G\preceq [T_-,g,F_+][E,1,1]_G \text{.} \end{align*}$$

More generally, if E is any elementary d-ary forest with the same number of roots as $F_+$ , then all the vertices of the form $[T_-,g,F_+][E',1,1]_G$ , for $E'$ obtained from E by replacing some number of d-carets with trivial trees, span the $1$ -skeleton of a cube in $\mathscr {X}_d(G_*)$ , and we declare that there is a cube in $\mathscr {X}_d(G_*)$ with this $1$ -skeleton. All of this is well defined up to the equivalence relation. Any nonempty intersection of cubes is a common face of each of them, so $\mathscr {X}_d(G_*)$ really is a cube complex. Within a given cube, the geometric realization of the finite subposet of $\mathscr {P}_d(G_*)$ given by the vertices of the cube is naturally a subdivision of the cube. Hence, there is a natural subdivision of $\mathscr {X}_d(G_*)$ that is identifiable with a subcomplex of $|\mathscr {P}_d(G_*)|$ .

A Morse function: Let

$$\begin{align*}h\, {\colon}\, \mathscr{X}_d(G_*)^{(0)}\to\mathbb{N} \end{align*}$$

be the function sending $[T_-,g,F_+]_G$ to the number of roots of $F_+$ , and call this the number of feet of this vertex. Note that adjacent vertices have distinct h values. By the construction of $\mathscr {X}_d(G_*)$ , it is clear that h can be extended to a map $h\, {\colon}\, \mathscr {X}_d(G_*)\to \mathbb {R}$ that restricts to an affine map on each cube. Hence, h satisfies all the requirements to be a discrete Morse function, in the sense of Bestvina of Brady [Reference Bashwinger and ZaremskyBB97]. In particular, we will be allowed to use discrete Morse theory when it comes up later. The function h also comes into play when proving that $\mathscr {X}_d(G_*)$ is contractible, which we do shortly.

Upward-local finiteness: The main advantage of $\mathscr {X}_d(G_*)$ over $|\mathscr {P}_d(G_*)|$ is that the former is what we will call “upward-locally finite,” as we now explain. Note that there are only finitely many elementary d-ary forests with a given number of roots. Also, note that for any $[1,g,1],[F,1,1]\in \mathscr {G}_d(G_*)$ such that the product $[1,g,1][F,1,1]$ makes sense, there exist $F'$ and $g'$ such that $[1,g,1][F,1,1]=[F',1,1][1,g',1]$ . More precisely, $F'$ and $g'$ are such that upon expanding we get $[1,g,1]=[F',g',F]$ , and then

$$\begin{align*}[1,g,1][F,1,1]=[F',g',F][F,1,1]=[F',g',1]=[F',1,1][1,g',1]\text{.} \end{align*}$$

In particular, given any vertex $x=[T_-,g,F_+]_G$ of $\mathscr {X}_d(G_*)$ , every vertex y with $x\preceq y$ is of the form $y=[T_-,g,F_+][E,1,1]_G$ for some elementary d-ary forest E. (Indeed, a priori y is of the form $[T_-,g,F_+][1,h,1][E,1,1]_G$ , but the above shows that we can ignore the $[1,h,1]$ factor for the purposes of characterizing y.) We conclude that for any x, there exist only finitely many y with $x\preceq y$ . We refer to this property by saying that $\mathscr {X}_d(G_*)$ is upward-locally finite.

Contractibility: To see that $\mathscr {X}_d(G_*)$ is contractible, we will show that the inclusion $\mathscr {X}_d(G_*)\to |\mathscr {P}_d(G_*)|$ is a homotopy equivalence. Using the usual notation of open and closed intervals in posets, every simplex of $|\mathscr {P}_d(G_*)|$ that is not in $\mathscr {X}_d(G_*)$ lies in the realization of a closed interval $[x,y]$ for $x=[T_-,g,F_+]_G\le y=[T_-,g,F_+][F,1,1]_G$ with F a nonelementary d-ary forest. Hence, it suffices to show that we can build up from $\mathscr {X}_d(G_*)$ to $|\mathscr {P}_d(G_*)|$ by gluing in the realizations of such closed intervals in some way that never changes the homotopy type. Using the above notation, the order we will use is in increasing order of the quantity $h(y)-h(x)$ . Thus, when we glue in $|[x,y]|$ , we do so along $|[x,y)|\cup |(x,y]|$ . This is the suspension of $|(x,y)|$ , so it suffices to see that $|(x,y)|$ is contractible. For this, we can mimic the proof of [Reference Witzel and ZaremskyWZ18, Lemma 4.7], which is the $d=2$ case. Recall that a poset $(Y, \sqsubseteq )$ is called conically contractible if there is a $y_0$ in Y and a poset map $g: Y \rightarrow Y$ such that $z \sqsubseteq g(z) \sqsupseteq y_0$ for all z in Y. A consequence of a poset being conically contractible is that its geometric realization is contractible. Since a poset and its opposite poset have isomorphic geometric realizations, we can also use the criterion $z \sqsupseteq g(z) \sqsubseteq y_0$ . See the discussion in [Reference QuillenQui78, Section 1.5] for more details. Now, given any $z \in (x,y]$ , define $g(z)$ to be the largest element of $[x,z]$ such that $x\preceq g(z)$ (the idea is to replace any d-ary forest with its unique largest elementary subforest). By our hypothesis, $g(z)$ is in $[x,y)$ and it is also clearly in $(x,y]$ , so we have $g(z)\in (x,y)$ . Let $y_0=g(y).$ Note that for any $z\in (x,y)$ , we have $g(z) \le y_0$ . Therefore, $z \ge g(z) \le y_0$ and $(x,y)$ is conically contractible.

Action and stabilizers: The group $\mathscr {T}_d(G_*)$ acts on $\mathscr {X}_d(G_*)$ by left multiplication, which is well defined since the equivalence relation defining vertices of $\mathscr {X}_d(G_*)$ and the partial order dictating which vertices span cubes are both given by right multiplication. Note too that h is invariant under this action. We claim that the stabilizer of a vertex with h value m is isomorphic to $G_m$ . Given such a vertex $[T_-,g,F_+]_G$ and an element $[U_-,h,U_+]$ of $\mathscr {T}_d(G_*)$ , we have $[U_-,h,U_+][T_-,g,F_+]_G=[T_-,g,F_+]_G$ if and only if $[F_+,g^{-1},T_-][U_-,h,U_+][T_-,g,F_+]=[1,h',1]$ for some $h'\in G_m$ (since $F_+$ has m roots, the “ $1$ ” here is the trivial forest with m roots, and $h'\in G_m$ ). Thus, $[U_-,h,U_+]\mapsto h'$ provides the desired isomorphism from the vertex stabilizer to $G_m$ . Now we claim that any cube stabilizer is a finite index subgroup of a vertex stabilizer. Consider a cube in $\mathscr {X}_d(G_*)$ , say with x its unique vertex with minimum h value and y its unique vertex with maximal h value. Since the action preserves h, the cube stabilizer lies in the stabilizer of the vertex x (and that of y). Since $\mathscr {X}_d(G_*)$ is upward-locally finite, $\operatorname {\mathrm {Symm}}(\{z\mid x\preceq z\})$ is finite, so the kernel of $\operatorname {\mathrm {Stab}}(x)\to \operatorname {\mathrm {Symm}}(\{z\mid x\preceq z\})$ has finite index in $\operatorname {\mathrm {Stab}}(x)$ . This kernel clearly fixes our cube pointwise, so we conclude that the stabilizer of the cube has finite index in the stabilizer of x.

Cocompactness: Let $\mathscr {X}_d(G_*)^{h\le m}$ be the full subcomplex of $\mathscr {X}_d(G_*)$ spanned by vertices with at most m feet. Since h is invariant under the action of $\mathscr {T}_d(G_*)$ , all the $\mathscr {X}_d(G_*)^{h\le m}$ are stabilized by this action. Given vertices $[T_-,g,F_+]_G$ and $[U_-,h,E_+]_G$ with the same h value, the product $[U_-,h,E_+][F_+,g^{-1},T_-]$ exists, lies in $\mathscr {T}_d(G_*)$ , and takes $[T_-,g,F_+]_G$ to $[U_-,h,E_+]_G$ . Hence, $\mathscr {T}_d(G_*)$ is transitive on vertices of a given h value. Since $\mathscr {X}_d(G_*)$ is upward-locally finite, and since the vertices of any given $\mathscr {X}_d(G_*)^{h\le m}$ only have finitely many h values, this shows that the action of $\mathscr {T}_d(G_*)$ on any $\mathscr {X}_d(G_*)^{h\le m}$ is cocompact.

Let us summarize all of the above.

4.2 Descending links

In this subsection and the next, we prove Theorem 4.1. In a now-standard approach, we will combine Proposition 4.2 with Brown’s Criterion and discrete Morse theory to reduce the problem to an analysis of descending links.

First, let us recall some background on discrete Morse theory and descending links in general. Let Y be an affine cell complex, in the sense of [Reference Bashwinger and ZaremskyBB97]. Let $h\, {\colon}\, Y\to \mathbb {R}$ be a map such that the image of the vertex set of Y is a closed, discrete subset of $\mathbb {R}$ . If the restriction of h to every positive-dimensional cell is a nonconstant affine map, then we call h a Morse function. These conditions ensure that for every cell, there is a unique vertex at which h achieves its maximum value on that cell. The descending star $\operatorname {st}^\downarrow \! v$ of a vertex v is the subcomplex of Y consisting of all cells for which v is this vertex with maximum h value. The descending link $\operatorname {lk}^\downarrow \! v$ of v is the link of v in $\operatorname {st}^\downarrow \! v$ , i.e., the space of directions out of v along which h strictly decreases. (One could analogously define ascending stars and links, but here we will only use the descending version.)

The point of Morse theory is that an understanding of the descending links can translate to an understanding of the whole complex. The following essentially combines the Morse Lemma from [Reference Bashwinger and ZaremskyBB97] with Brown’s Criterion from [Reference BrothierBro87], to produce a sufficient condition for a group to be of type $\operatorname {\mathrm {F}}_n$ . See [Reference SteinSZ21, Lemma 3.1] for an $\operatorname {\mathrm {F}}_\infty $ version of the following, which works exactly analogously for $\operatorname {\mathrm {F}}_n$ .

Now let us discuss descending links in $\mathscr {X}_d(G_*)$ . Since $\mathscr {X}_d(G_*)$ is a cube complex, the link of a vertex v is a simplicial complex, with a k-simplex for each $(k+1)$ -cube containing v. The descending link is the subcomplex whose simplices correspond to cubes for which the vertex is the one with maximum h value, i.e., the most feet. Thus, a k-simplex in $\operatorname {lk}^\downarrow \! v$ , say with $h(v)=m$ , is represented by $v[1,g,E]_G$ , for g some element of $G_m$ and E some elementary d-ary forest with m leaves and $k+1$ carets. The face relation is given by removing carets from E. Note that the barycentric subdivision of $\operatorname {lk}^\downarrow \! v$ is naturally a subcomplex of $|\mathscr {P}_d(G_*)|$ , namely the full subcomplex spanned by all the $v[1,g,E]_G$ .

There is a convenient model for the descending links. Note that, up to the action of $\mathscr {T}_d(G_*)$ , $\operatorname {lk}^\downarrow \! v$ only depends on $h(v)$ . Thus, the following complex is isomorphic to every descending link of a vertex with m feet:

The culmination of Sections 4.1 and 4.2 is the following, which holds for any d-ary cloning system.

4.3 Descending links in the braided case

Now we return to the special case from Section 3.3. We have a braided self-similar group $G\le \operatorname {br}\!\operatorname {\mathrm {Aut}}(\mathcal {T}_d)$ of type $\operatorname {\mathrm {F}}_n$ , and need to prove that $\operatorname {br}\!V_d(G)=\mathscr {T}_d(B_*\wr G)$ is of type $\operatorname {\mathrm {F}}_n$ . As indicated by Proposition 4.5, the key thing to show is that $\mathscr {L}_d^m(B_*\wr G)$ is $(n-1)$ -connected for all but finitely many m. To analyze the higher connectivity of $\mathscr {L}_d^m(B_*\wr G)$ , we will use a strategy that essentially comes from [Reference Benli, Grigorchuk and de la HarpeBFM+16] in the case when $G=\{1\}$ and $d=2$ , and more generally from [Reference Skipper and ZaremskySW] in the case when G is braided self-identical.

The idea is to map $\mathscr {L}_d^m(B_*\wr G)$ to a complex that is easier to understand. We will use the following complex, considered in [Reference Skipper and ZaremskySW].

Our next goal is to map $\mathscr {L}_d^m(B_*\wr G)$ to ${\mathbb {D}}_d(\mathcal {S}_{1,m}^0)$ in a certain way. First, we describe a useful alternate viewpoint of elementary d-ary forests. Let $L_{m-1}$ be the linear graph with m vertices, that is, the graph with m vertices labeled $1$ to m, and $m-1$ edges, one connecting i to $i+1$ for each $1\le i<m$ . Call a subgraph of $L_{m-1}$ a d-matching on $L_{m-1}$ if each of its connected components is a subgraph of length $d-1$ . The set of d-matchings forms a simplicial complex called the d-matching complex, denoted by ${\mathcal {M}}_d(L_{m-1})$ , where a matching forms a k-simplex whenever it consists of $k+1$ disjoint paths, and the face relation is given by inclusion. Observe that there is a bijection between the set of elementary d-ary forests with m leaves and the set of d-matchings on $L_{m-1}$ . Under the identification, d-carets correspond to paths of length $d-1$ . See Figure 5 for example.

Figure 5:

An example of the bijective correspondence between elementary $3$ -ary forests with nine leaves and simplices of ${\mathcal {M}}_3(L_{8})$ .

Let $\mathcal {S}_m=\mathcal {S}_{1,m}^0$ be the unit disk with m marked points given by fixing an embedding $L_{m-1} \hookrightarrow \mathcal {S}_m$ of the linear graph with $m-1$ edges into $\mathcal {S}_{1,m}^0$ (so the marked points are the images of the vertices). The braid group $B_m$ on m strands is isomorphic to the mapping class group of the disk with m marked points [Reference BirmanBir75], so we have an action of $B_m$ on ${\mathbb {D}}_{d}(\mathcal {S}_m)$ , which will be convenient to take to be a right action.

Define a map $\mu \, {\colon}\, \mathscr {L}_d^m(B_*\wr G) \rightarrow {\mathbb {D}}_{d}(\mathcal {S}_m)$ as follows. For $[1, \beta (f_1, \dots , f_m), E]_G$ a simplex in $\mathscr {L}_d^m(B_*\wr G)$ , first view the elementary d-ary forest E as a d-matching in ${\mathcal {M}}_d(L_{m-1})$ as above. Then take a disk tubular neighborhood of the d-matching, call it $D_E$ , to obtain a simplex in ${\mathbb {D}}_{d}(\mathcal {S}_m)$ . Forget the labels $(f_1, \dots , f_m)$ , and then act on $D_E$ by applying $\beta ^{-1}$ on the right. In this way, we obtain a simplicial map

$$ \begin{align*} \mu \, {\colon}\, \mathscr{L}_d^m(B_*\wr G) & \to {\mathbb {D}}_{d}(\mathcal{S}_m) \\ [1, \beta (f_1, \dots, f_m), E]_G & \mapsto (D_E)\beta^{-1} \text{ .} \end{align*} $$

Proof Say E has q roots. Let $\gamma (g_1,\dots ,g_q)\in B_q\wr G$ , so

$$\begin{align*}[1, \beta (f_1, \dots, f_m), E]_G=[1,\beta(f_1, \dots, f_m), E][1,\gamma(g_1,\dots,g_q),1]_G \text{.} \end{align*}$$

Let $\gamma '(g_1',\dots ,g_m')\in B_q\wr G$ and $E'$ be such that

$$\begin{align*}[1,1,E][1,\gamma(g_1,\dots,g_q),1] = [1,\gamma'(g_1',\dots,g_m'),1][1,1,E'] \text{.} \end{align*}$$

Then

$$\begin{align*}[1,\beta(f_1, \dots, f_m), E][1,\gamma(g_1,\dots,g_q),1]_G = [1,\beta\gamma'(f_1', \dots, f_m')(g_1',\dots,g_m'),1][1,1,E']_G\text{,} \end{align*}$$

for some $f_i'$ , so $\mu $ sends this to $(D_{E'})(\beta \gamma ')^{-1}$ . We need to show that this equals $(D_E)\beta ^{-1}$ , or equivalently that $(D_{E'})(\gamma ')^{-1}=D_E$ , i.e., $(D_E)\gamma '=D_{E'}$ . Note that $\gamma '$ is obtained by taking $\gamma $ , turning each strand corresponding to a root of E with a d-caret on it into d parallel strands (call this intermediate braid $\gamma "$ ), and then on each such collection of d parallel strands applying some $\phi (g_i)\in B_d$ to it (where $\phi $ is as in Section 2.1). When acting on $D_E$ though, since each of these local copies of the $\phi (g_i)$ is supported in the interior of one of the disks in the disk system, they do not change the (equivalence class of the) disk system. It is clear that $(D_E)\gamma "=D_{E'}$ , so we conclude that also $(D_E)\gamma '=D_{E'}$ .

One can visualize $\mu $ as taking $[1,\beta (f_1,\dots ,f_n),E]_G$ , viewing it as the braid $\beta $ with strands labeled by the $f_i$ at the bottom and with E serving to “merge” strands together, and then forgetting the labels $f_i$ , considering the merges as d-matchings, enlarging them to disks, “combing straight” the braid, and seeing where the disks are taken. See Figure 6 for example. Note that the resulting simplex $(D_E)\beta ^{-1}$ of ${\mathbb {D}}_{d}(\mathcal {S}_m)$ has the same dimension as the simplex $[1, \beta (f_1, \dots , f_m), E]_G$ of $\mathscr {L}_d^m(B_*\wr G)$ , so in particular $\mu $ is simplexwise injective. Also, note that if $d>2$ , then the $\phi (g_i)$ from the proof of Lemma 4.7 could change the d-matchings, but cannot change their corresponding disks, hence why we use disk complexes instead of, say, arc complexes.

Figure 6:

An illustration of $\mu \, {\colon}\, \mathscr {L}_3^6(B_*\wr G) \rightarrow {\mathbb {D}}_3(\mathcal {S}_6)$ . The successive pictures show the process of flattening to a $3$ -matching, expanding the matching to disks, forgetting the labels, and “combing straight” the braid.

In the following proofs, we will need a useful tool introduced by Hatcher and Wahl called the complete join [Reference Hatcher and WahlHW10].

The main result regarding complete joins that we will use is the following.

Now we would like to use the map $\mu \, {\colon}\, \mathscr {L}_d^m(B_*\wr G) \to {\mathbb {D}}_{d}(\mathcal {S}_{1,m}^0)$ to prove higher connectivity of $\mathscr {L}_d^m(B_*\wr G)$ , and hence of the descending links in the Stein–Farley complex. First, we need the following, which is proved in [Reference Skipper and ZaremskySW].

Now we can prove Theorem 4.1.

5.1 Polysimplicial structure

Now let us describe the polysimplicial structure on $\mathscr {X}_Z$ . A polysimplex is a euclidean polytope that is a product of simplices. A polysimplicial complex is an affine cell complex whose cells are polysimplices, and such that any two cells intersect in a (possibly empty) common face of each. Polysimplicial complexes are a simultaneous generalization of simplicial and cubical complexes. The polysimplicial structure on $\mathscr {X}_Z$ is built out of the following pieces.

Definition 5.4 (Basic polysimplex, bottom vertex)

Let $[\tau ]_Z$ be a vertex in $\mathscr {X}_Z$ , say with $h([\tau ]_Z)=m$ (here h is the “number of feet” function from before). For each ${1\le i\le m}$ , let $S_i$ be one of the following sets:

$$\begin{align*}\{1\}\text{, } \{1,\wedge(a^k,1)\}\text{ for some }k\in\mathbb{Z}\text{, } \{1,\wedge^2\}\text{, or }\{1,\wedge(a^k,1),\wedge^2\}\text{ for some }k\in\mathbb{Z}\text{.} \end{align*}$$

Then the corresponding basic polysimplex $\operatorname {\mathrm {poly}}([\tau ];S_1,\dots ,S_m)$ is the full subcomplex of $\mathscr {X}_Z$ spanned by the vertex set

$$\begin{align*}\{[\tau](\Delta_1\oplus\cdots\oplus\Delta_m)_Z\mid \Delta_i\in S_i\}\text{.} \end{align*}$$

The bottom vertex of this basic polysimplex is $[\tau ]_Z$ .

Note that $\operatorname {\mathrm {poly}}([\tau ];S_1,\dots ,S_m)$ is a subdivision of a product of simplices, one for each $S_i$ , where the dimension of the simplex corresponding to $S_i$ is $|S_i|-1$ . In particular, it makes sense to view this as a polysimplex.

It is clear that every weakly elementary simplex lies in some basic polysimplex, so the basic polysimplices cover $\mathscr {X}_Z$ . To see that the basic polysimplices form a polysimplicial complex with $\mathscr {X}_Z$ as a simplicial subdivision, it remains to show that any intersection of two basic polysimplices is a common face of each. Before proving this, we need a definition.

The terminology “nonexpanding” is inspired by the nonbraided case, as in [Reference Brady, Burillo, Cleary and SteinBM16], where groupoid elements are homeomorphisms between disjoint unions of copies of the Cantor set.

Proof The proof of Lemma 5.3 of [Reference Brady, Burillo, Cleary and SteinBM16] for the Röver group can almost be copied verbatim for the braided case (up to changing notation and some left/right conventions), with just one step requiring justification, which we will point out when we reach it. Let $P=\operatorname {\mathrm {poly}}([\tau ];S_1,\dots ,S_m)$ and $Q=\operatorname {\mathrm {poly}}([\omega ];T_1,\dots ,T_q)$ be basic polysimplices. Each $S_i$ has a natural total order, consistent with $\le $ in $\mathscr {P}_Z$ when considered up to the “mod Z” equivalence relation, so it makes sense to take the minimum of a collection of elements of a given $S_i$ . Define a binary operation $\wedge _P$ on the vertices of P via

Define $\wedge _Q$ analogously. If $P\cap Q=\emptyset $ , we are done, so suppose not. Let $v,v'\in P\cap Q$ , and we claim $v\wedge _P v'=v\wedge _Q v'$ . Without loss of generality, $v\wedge _P v'=[\tau ]_Z$ and $v\wedge _Q v'=[\omega ]_Z$ . Choose $\Delta _i,\Delta _i'\in S_i$ and $\Omega _i,\Omega _i'\in T_i$ such that

$$ \begin{align*} &v=[\tau](\Delta_1\oplus\cdots\oplus\Delta_m)_Z=[\omega](\Omega_1\oplus\cdots\oplus\Omega_q)_Z\\ \text{ and } &v'=[\tau](\Delta_1'\oplus\cdots\oplus\Delta_m')_Z=[\omega](\Omega_1'\oplus\cdots\oplus\Omega_q')_Z \text{.} \end{align*} $$

Now, for appropriate r and p, choose $\beta \in B_r$ , $\beta '\in B_p$ , and $z_1,\dots ,z_r,z_1',\dots ,z_p'\in Z$ such that

$$ \begin{align*} &[\tau](\Delta_1\oplus\cdots\oplus\Delta_m)=[\omega](\Omega_1\oplus\cdots\oplus\Omega_q)\beta(z_1,\dots,z_r)\\ \text{ and } &[\tau](\Delta_1'\oplus\cdots\oplus\Delta_m')=[\omega](\Omega_1'\oplus\cdots\oplus\Omega_q')\beta(z_1',\dots,z_p') \text{.} \end{align*} $$

(As before, we identify $B_r\wr Z$ and $B_p\wr Z$ with their images in $\mathscr {G}$ under $g\mapsto [1,g,1]$ .) Solving for $[\omega ]^{-1}[\tau ]$ in each equation yields

$$ \begin{align*} [\omega]^{-1}[\tau] &= (\Omega_1\oplus\cdots\oplus\Omega_q)\beta(z_1,\dots,z_r)(\Delta_1\oplus\cdots\oplus\Delta_m)^{-1} \\ &=(\Omega_1'\oplus\cdots\oplus\Omega_q')\beta'(z_1',\dots,z_p')(\Delta_1'\oplus\cdots\oplus\Delta_m')^{-1} \text{.} \end{align*} $$

We want to conclude that $[\tau ]_Z=[\omega ]_Z$ , and this is the one step that cannot be copied verbatim from the nonbraided case. However, it does work analogously, using the above definition of nonexpanding. Since $v\wedge _P v'=[\tau ]_Z$ , we either have $\Delta _i=1$ or $\Delta _i'=1$ for each i. This shows that $[\omega ]^{-1}[\tau ]$ is nonexpanding. Analogously, $[\tau ]^{-1}[\omega ]$ is nonexpanding. The only way this can happen, given the options for the $\Delta _i$ and $\Omega _i$ , is if every $\Delta _i$ and $\Omega _i$ is trivial. We conclude that $[\tau ]_Z=[\omega ]_Z$ , so $\wedge _P$ and $\wedge _Q$ agree when restricted to $P\cap Q$ . By the exact same argument as in the proof of [Reference Brady, Burillo, Cleary and SteinBM16, Lemma 5.3], this shows that without loss of generality, $[\tau ]=[\omega ]$ , and $P\cap Q=\operatorname {\mathrm {poly}}([\tau ];S_1\cap T_1,\dots ,S_m\cap T_m)$ , which is a common face of both P and Q.

At this point, we can view $\mathscr {X}_Z$ either with its simplicial structure, or with this new polysimplicial structure. We will write $\mathscr {X}_Z$ for both, and no confusion should arise.