Orderings of Witzel-Zaremsky-Thompson groups

TL;DR

This paper proves that the Witzel-Zaremsky-Thompson group is orderable when constructed from a compatible direct system of orderable groups, expanding understanding of group orderability properties.

Contribution

It establishes the orderability of the Witzel-Zaremsky-Thompson group under specific conditions, a new result in the study of these groups.

Findings

Witzel-Zaremsky-Thompson group is orderable under certain conditions

Orderability depends on compatibility of the direct system of groups

Advances understanding of algebraic properties of complex groups

Abstract

We prove the orderability of the Witzel-Zaremsky-Thompson group for a direct system of orderable groups under a certain compatibility assumption.

Equations

$$\varsigma_{n}^{k}(g)m=\left\{\begin{array}[]{ll}gm&\text{if }m\leq k,gm\leq gk,\\ gm+1&\text{if }m<k,gm>gk,\\ g(m-1)&\text{if }m>k,g(m-1)<gk,\\ g(m-1)+1&\text{if }m>k,g(m-1)\geq gk.\end{array}\right.$$

$$\Pi =\{[T_{-},g,T_{+}]\in \mathcal{T}(G_{\ast });g\in G_n,1{<}_{n}g\}⨿\{f\in F;1{<}_{F}f\}.$$

$$\kappa_{n}^{k}(\sigma_{i})=\left\{\begin{array}[]{ll}\sigma_{i+1}&\text{if }k<i,\\ \sigma_{i+1}\sigma_{i}&\text{if }k=i,\\ \sigma_{i}\sigma_{i+1}&\text{if }k=i+1,\\ \sigma_{i}&\text{if }i+1<k,\end{array}\right.$$

Full text

1. Introduction

Thompson’s groups are interesting countable groups and several kinds of their generalizations have been developed and studied. As one of such examples, Witzel and Zaremsky introduced the group $\mathscr{T}({G_{\ast}})$ associated to a direct system $(G_{n})$ of groups satisfying certain axioms called the cloning system [6]. We call it the Witzel-Zaremsky-Thompson group for $(G_{n})$.

The Witzel-Zaremsky-Thompson group $\mathscr{T}({G_{\ast}})$ can be considered as not only a generalization of Thompson’s groups but also a limit of the groups $G_{n}$. We prove the group $\mathscr{T}({G_{\ast}})$ inherits the orderability of the groups $G_{n}$ under the assumption that there exists an ordering of $G_{n}$ which are compatible with the cloning maps. A precise definition of the compatibility is given in Section 2.

Proposition 1.1**.**

Let $(G_{n})$ be a cloning system.

1. (i)

The Witzel-Zaremsky-Thompson group $\mathscr{T}({G_{\ast}})$ for $(G_{n})$ is left-orderable if the groups $G_{n}$ admit left-orderings which are compatible with the cloning maps. 2. (ii)

Suppose that the cloning system is pure. The group $\mathscr{T}({G_{\ast}})$ is bi-orderable if the groups $G_{n}$ admit bi-orderings which are compatible with the cloning maps.

The statement (ii) in Proposition 1.1 is proved by an argument which is a straightforward generalization of that used in [3]. The author does not know whether the assumption that the cloning system is pure is essential or not.

2. Preliminaries

Let $(G_{n})$ be a direct system of groups. That is, an injective homomorphism $G_{n}\hookrightarrow G_{n+1}$ is given for each $n\in\mathbb{N}$. The Witzel-Zaremsky-Thompson group $\mathscr{T}({G_{\ast}})$ is simply a certain subgroup of the cancellative group of the Brin-Zappa-Szép product $G\bowtie\mathscr{F}$, where $G$ is the direct limit of the system $(G_{n})$ and $\mathscr{F}$ is the monoid of binary forests. However we recall the definition of $\mathscr{T}({G_{\ast}})$ in a fashion describing its elements.

Let $\mathfrak{S}_{n}$ be the symmetric group on the set $\{1,\dots,n\}$. Set the map $\varsigma_{n}^{k}\colon\mathfrak{S}_{n}\to\mathfrak{S}_{n+1}$ by

[TABLE]

Definition 2.1**.**

Suppose that a group homomorphism $\rho_{n}\colon G_{n}\to\mathfrak{S}_{n}$ and an injective map $\kappa_{n}^{k}\colon G_{n}\to G_{n+1}$ is given for each $n\in\mathbb{N}$ and for each $k=1,\dots,n$.

The triple $(G_{n})=((G_{n}),(\rho_{n}),(\kappa_{n}^{k}))$ is a cloning system if the following three axioms are satisfied.

1. (1)

$\kappa_{n}^{k}(gh)=\kappa_{n}^{k}(g)\kappa_{n}^{\rho_{n}(g)k}(h)$ for $g,h\in G_{n}$, 2. (2)

$\kappa_{n+1}^{k}\circ\kappa_{n}^{l}=\kappa_{n+1}^{l+1}\circ\kappa_{n}^{k}$ for $1\leq k<l\leq n$, and 3. (3)

$\rho_{n}(\kappa_{n}^{k}(g))=\varsigma_{n}^{k}(\rho_{n}(g))\text{ or }s_{k}\varsigma_{n}^{k}(\rho_{n}(g))$ for $g\in G_{n}$ and for $1\leq k\leq n$, where the symbol $s_{k}$ means the transposition $(k,k+1)\in\mathfrak{S}_{n+1}$.

The maps $\kappa_{n}^{k}\colon G_{n}\to G_{n+1}$ are called the cloning maps of the cloning system $(G_{n})$.

Suppose that a cloning system $(G_{n})=((G_{n}),(\rho_{n}),(\kappa_{n}^{k}))$ is given. A tree diagram is a triple $(T_{-},g,T_{+})$, where $g\in G_{n}$ for certain $n\in\mathbb{N}$ and $T_{\pm}$ are rooted planar binary trees with $n$ leaves. A simple expansion of $(T_{-},g,T_{+})$ is a tree diagram of the form $(T^{\prime}_{-},\kappa_{n}^{k}(g),T^{\prime}_{+})$. Here, $T^{\prime}_{-}$ and $T^{\prime}_{+}$ are trees obtained by adjoining carets to the $k$-th and $(\rho(g)k)$-th leaves of $T_{-}$ and $T_{+}$, respectively. An expansion of $(T_{-},g,T_{+})$ is a tree diagram obtained as an iterated simple expansions. Two tree diagrams are equivalent if they have a common expansion. The equivalence class represented by a tree diagram $(T_{-},g,T_{+})$ will be denoted by $[T_{-},g,T_{+}]$. The Witzel-Zaremsky-Thompson group $\mathscr{T}({G_{\ast}})$ consists of equivalence classes of tree diagrams as a set. For two elements $[S_{-},f,S_{+}],[T_{-},g,T_{+}]\in\mathscr{T}({G_{\ast}})$, their product is defined as follows. There always exist tree diagrams $(S^{\prime}_{-},f^{\prime},S^{\prime}_{+})$ and $(T^{\prime}_{-},g^{\prime},T^{\prime}_{+})$ representing $[S_{-},f,S_{+}]$ and $[T_{-},g,T_{+}]$, respectively and such that $S^{\prime}_{+}=T^{\prime}_{-}$. The product $[S_{-},f,S_{+}][T_{-},g,T_{+}]$ is defined to be $[S^{\prime}_{-},f^{\prime}g^{\prime},T^{\prime}_{+}]$. This is well-defined. The cloning system $(G_{n})$ is pure if the homomorphism $\rho_{n}\colon G_{n}\to\mathfrak{S}_{n}$ is trivial for each $n\in{\mathbb{N}}$.

The subgroup of $\mathscr{T}({G_{\ast}})$ consisting of elements represented by the form $(T_{-},1,T_{+})$ is isomorphic to the Thompson’s group $F$. and we denote this subgroup again by $F$. If we denote by $P\mathscr{T}({G_{\ast}})$ the kernel of natural projection $p\colon\mathscr{T}({G_{\ast}})\to V$, then the group $P\mathscr{T}({G_{\ast}})$ consists of the elements of the form $[T,g,T]$, where $g\in{\rm Ker}\rho_{n}$ and $n$ is the number of leaves of $T$. If the cloning system $(G_{n})$ is pure, then the image of the projection $p\colon\mathscr{T}({G_{\ast}})\to V$ is equal to Thompson’s group $F<V$ and thus there exists an isomorphism $\mathscr{T}({G_{\ast}})\cong P\mathscr{T}({G_{\ast}})\rtimes F$.

When we fix a left-ordering or a bi-ordering $<_{n}$ of $G_{n}$ for each $n$, we say the orderings are compatible with the cloning maps $\kappa_{n}^{k}$’s if $1<_{n+1}\kappa_{n}^{k}(g)$ for any $n$ and any $k$ whenever $1<_{n}g$.

3. Proof of the main proposition

Proof of Proposition 1.1.

Let $(G_{n})=((G_{n}),(\rho_{n}),(\kappa_{n}^{k}))$ be a cloning system. Suppose that a left-ordering $<_{n}$ of the group $G_{n}$ is given for each $n$ and the orderings $<_{n}$ are compatible with the cloning maps. Since the Thompson’s group $F$ is known to be bi-orderable, we fix a left-ordering $<_{F}$ of $F$. Set the subset $\Pi$ of the Witzel-Zaremsky-Thompson group $\mathscr{T}({G_{\ast}})$ for $(G_{n})$ by

[TABLE]

Since the orders $<_{n}$ are compatible with the cloning maps, the set $\Pi$ is well-defined. Further it is easy to verify that $\Pi$ is a positive cone and thus defines a left-ordering of $\mathscr{T}({G_{\ast}})$. This completes the proof of the statement (i).

If the groups $G_{n}$ admit bi-orderings which are compatible with the cloning maps, then the bi-ordering on $P\mathscr{T}({G_{\ast}})$ is induced. Under the assumption that the cloning system $(G_{n})$ is pure, since both the groups $P\mathscr{T}({G_{\ast}})$ and $F$ are bi-orderable, the group $\mathscr{T}({G_{\ast}})$ which is isomorphic to the semi-direct product $P\mathscr{T}({G_{\ast}})\rtimes F$ is also. This completes the proof of the statement (ii). ∎

Remark 3.1*.*

If we fix a left-ordering or bi-ordering on the group $\mathscr{T}({G_{\ast}})$, its restriction on $P\mathscr{T}({G_{\ast}})$ uniquely determines a family of left-ordering or bi-ordering respectively on the groups $G_{n}$. In fact, if for each rooted planar binary tree $T$ with $n$ leaves, we denote by $G_{T}$ the subgroup of $P\mathscr{T}({G_{\ast}})$ consisting of elements represented by tree diagrams of the form $(T,g,T)$, where $g\in G_{n}$, then $G_{n}$ is isomorphic to $G_{n}$. Furthermore, if $T$ and $T^{\prime}$ are rooted planar binary trees with same number of leaves, then in $\mathscr{T}({G_{\ast}})$ the subgroups group $G_{T}$ and $G^{\prime}_{T}$ are conjugate to each other by an element of $F$. Thus if we assume the cloning system $(G_{n})$ is pure, the converse of each statement of Proposition 1.1 always holds.

4. Examples

In this section we discuss examples of Witzel-Zaremsky-Thompson groups to which Proposition 1.1 is applicable.

4.1. The braided Thompson group

Let $B_{n}$ be the braid group of $n$ strands and $\sigma_{1},\dots,\sigma_{n-1}\in B_{n}$ the Artin generators. Taking $\rho_{n}\colon B_{n}\to\mathfrak{S}_{n}$ to be the natural projection and setting $\kappa_{n}^{k}\colon B_{n}\to B_{n+1}$ by

[TABLE]

we have a well-defined cloning system on $(B_{n})$. Then the Witzel-Zaremsky-Thompson group $\mathscr{T}({B_{n}})$ for $(B_{n})$ is isomorphic to the group called the braided Thompson group $BV$ which was independently introduced by Brin and Dehornoy [2][4].

Recall that $\beta\in B_{n}$ is positive with respect to the Dehornoy ordering if and only if $\beta$ is represented by a word $w$ in the Artin generators which satisfies the following condition (D):

(D) $\sigma_{i}$ occurs in $w$ but $\sigma_{1}^{\pm 1},\dots,\sigma_{i-1}^{\pm 1},\sigma_{i}^{-1}$ does not for certain $i$.

It is not difficult to verify only by definition of the cloning maps that if $w$ is a representation of $\beta$ satisfying the condition (D) then $\kappa_{n}^{k}(w)$ also. Hence the following lemma holds:

Lemma 4.1**.**

The Dehornoy orderings of the braid groups $B_{n}$ are compatible with the cloning maps of $(B_{n})$ defined above.

By Proposition 1.1 and Lemma 4.1, we have in easy way the following theorem which was first proved in [4].

Theorem 4.2**.**

The braided Thompson group $BV$ is left-orderable.

4.2. The pure braided Thompson group

Let $P_{n}$ be the pure braid group of $n$ strands. We also denote the restriction on $P_{n}$ of the map $\kappa_{n}^{k}\colon B_{n}\to\mathfrak{S}_{n}$ we set in the previous subsection again by $\kappa_{n}^{k}$. Take $\rho_{n}\colon P_{n}\to\mathfrak{S}_{n}$ to be the trivial homomorphism. Then we have a well-defined cloning system on $(P_{n})$ and the Witzel-Zaremsky-Thompson group $\mathscr{T}({P_{n}})$ for $(P_{n})$ which is isomorphic to the pure braided Thompson group $BF$ introduced by Brady, Burillo, Cleary and Stein in [1].

Lemma 4.3** ([3]).**

The orderings on the braid groups $P_{n}$ induced from the Magnus orderings of the free groups via the Artin combing are compatible with the cloning maps on $(P_{n})$ defined above.

Theorem 4.4** ([3]).**

The pure braided Thompson group $BF$ is bi-orderable.

4.3. Witzel-Zaremsky-Thompson group for direct powers of a group

For arbitrary group $G$, set $G_{n}$ to be the $n$-th direct power $G^{n}$ of $G$. Fix injective homomorphisms $\phi_{1},\phi_{2}$ of $G$. If we define $\kappa_{n}^{k}\colon G^{n}\to G^{n+1}$ by $\kappa_{n}^{k}(g_{1},\dots,g_{n})=(g_{1},\dots,\phi_{1}(g_{k}),\phi_{2}(g_{k}),\dots,g_{n})$ and set $\rho_{n}\colon G^{n}\to\mathfrak{S}_{n}$ to be the trivial homomorphism, then we have a cloning system on direct powers $(G^{n})$ of the group $G$ and the Witzel-Zaremsky-Thompson group $\mathscr{T}({G^{n}})$ which was introduced by Tanushevski [5]. Suppose that $G$ is left-orderable and fix a left-ordering $<$ of $G$. If the homomorphism $\phi_{1}\colon G\to G$ preserves the order of $G$, then the lexicographic orderings of $G^{n}$’s induced by $<$ are compatible with the cloning maps of $(G^{n})$. Further if the order on $G$ is bi-invariant, then the induced orders on $G^{n}$ are also. Since the identity map of $G$ trivially preserves all the orderings of $G$, by Proposition 1.1 we have the following Theorem:

Theorem 4.5**.**

If the group $G$ is left-orderable or bi-orderable, then the Witzel-Zaremsky-Thompson group $\mathscr{T}({G^{n}})$ of the direct powers of $G$ is also left-orderable or bi-orderable, respectively.

Acknowledgments. The author is grateful to the referee for carefully reading the manuscript and pointing out mistake in it. The author was supported by JSPS Research Fellowships for Young Scientists (26$\cdot$110).