Quasi-Automorphism Groups of Type F-infinity

TL;DR

This paper proves that the quasi-automorphism groups of the infinite binary tree, including QF, QT, and QV, have type F-infinity, using hybrid diagrams and actions on CAT(0) cubical complexes.

Contribution

It introduces a novel use of hybrid diagrams to establish the finiteness properties of these quasi-automorphism groups.

Findings

All groups studied have type F-infinity.

Hybrid diagrams share properties with planar and braided diagrams.

Groups act properly on CAT(0) cubical complexes.

Abstract

The groups QF, QT, and QV are groups of quasi-automorphisms of the infinite binary tree. Their names indicate a similarity with Thompson's well-known groups F, T, and V. We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type F-infinity. Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on CAT(0) cubical complexes.

Equations

$${\Delta }_1\sim {\Delta }_2\iff {\Delta }_1\circ \Pi ={\Delta }_2,$$

$$\tilde{\Delta }\cdot (\Delta ,\Psi )=(r(\tilde{\Delta }\circ \Delta ),\Psi ).$$

$$S_1=\{j\in \{1,\dots ,n\}\mid \text{the bottom of }{w}_j\text{ is attached to a transistor of }{\Delta }_1\}$$

$$S_2=\{j\in \{1,\dots ,n\}\mid \text{the bottom of }{\hat{w}}_j\text{ is attached to a transistor of }{\Delta }_2\}.$$

$$D_v=\{(m,n,p)\mid m,n\in \{1,\dots ,k\};p\in \{1,\dots ,\ell \},m\ne n\},$$

$$A_v=\{(m)\mid m\in \{1,\dots ,k\}\}.$$

$$S=\{(m_1,n_1,p_1),\dots ,(m_{\alpha },n_{\alpha },p_{\alpha })\}\cup \{(m_1^{\prime }),\dots ,(m_{\beta }^{\prime })\}$$

$$m_1,n_1,\dots ,m_{\alpha },n_{\alpha },m_1^{\prime },\dots ,m_{\beta }^{\prime }$$

$$p_1,\dots ,p_{\alpha }.$$

$$h(010)=\sigma (0)10=0110.$$

$$\{{\Psi }_1,\dots ,{\Psi }_m\}$$

$$X_n=X_{n-1}\bigcup _{{\coprod }_v{\mathrm{lk}}_{\downarrow }(v)}\left(\coprod _v{C}_v\right),$$

$$\{(m,m+1,p)\mid m\in \{1,\dots ,k-1\};p\in \{1,\dots ,\ell \}\},$$

$$\{(m,m+1,p)\mid m\in \{1,\dots ,k-1\};p\in \{1,\dots ,\ell \}\}\bigcup \{(k,1,p)\mid p\in \{1,\dots ,\ell \}\},$$

$$(1,2,\alpha ),(3,4,\beta ),(1,2,\gamma )$$

$$(2,3,\beta ),(4,5,\gamma ),(1,2,\alpha ).$$

$$\mathcal{C}=\{(1,2,\alpha ),(2,3,\alpha )\mid \alpha \in \{1,\dots ,\ell \}\}.$$

$$\hat{\mathcal{C}}=\{C(m,m+1,\beta )\mid m\in \{1,2\};\beta \in \{1,\dots ,\ell \}\}.$$

$$C(m_1,m_1+1,{\beta }_1),\dots ,C(m_{n+3},m_{n+3}+1,{\beta }_{n+3})$$

$$\bigcap _{i=1}^{n+3}C(m_i,m_{i+1},{\beta }_i)$$

$$\bigcap _{i=1}^{t}C(m_i,m_i+1,{\beta }_i),$$

$$\{(m,m+1,\gamma )\mid m\in \{3,\dots ,k-1\};\gamma \in (\{1,\dots ,\ell \}-\{{\beta }_1,\dots ,{\beta }_i\})\}$$

$$\{(m,m+1,\gamma )\mid m\in \{4,\dots ,k-1\};\gamma \in (\{1,\dots ,\ell \}-\{{\beta }_1,\dots ,{\beta }_i\})\}.$$

$$k^{\prime }\ge k-3\ge 3(n+1)+5-3=3n+5\ge 3(n-t+2)+5$$

$${\ell }^{\prime }\ge \ell -t\ge 2n+5-t.$$

$$\bigcap _{v\in S}\mathrm{lk}(v)$$

$$\bigcap _{v\in S}\mathrm{lk}(v)\cong {\mathrm{lk}}_{\downarrow }^{QV}(x^{k^{\prime }}{a}^{{\ell }^{\prime }}),$$

$$4n+9-2|S|\ge 4(n+1-|S|)+5$$

$$2n+5-|S|\ge 2(n+1-|S|)+3.$$

Full text

We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type $F_{\infty}$. Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on CAT([math]) cubical complexes.

1. Introduction

Let $\Gamma$ be a graph. A quasi-automorphism of $\Gamma$ is a bijection of the vertices that preserves adjacency, with at most finitely many exceptions. Following the notation of [14], we will let $QV$ denote the group of quasi-automorphisms $h$ of the infinite binary tree that also take the left and right children of a given vertex $v$ to the left and right children of $h(v)$, again with at most finitely many exceptions. The notation “$QV$” indicates that $QV$ is a collection of quasi-automorphisms that bears a family resemblance to Thompson’s group $V$. (A standard reference for Thompson’s groups is [6]. We will assume a basic familiarity with that source or its equivalent throughout this article.)

Groups of quasi-automorphisms have been the subject of several recent studies. Lehnert conjectured in his thesis that the group $QV$ is a universal group with context-free coword problem; i.e., a universal $coCF$ group. Bleak, Matucci, and Neunhöffer [2] have produced an embedding of $QV$ into Thompson’s group $V$, and thus proved that Lehnert’s conjecture is equivalent to the conjecture that $V$ is itself a universal $coCF$ group. More recently, Nucinkis and St. John-Green [14] have studied the finiteness properties of $QV$ and related groups. They introduced additional groups $QF$, $QT$, $\bar{Q}T$, and $\bar{Q}V$. The groups $QF$ and $QT$ are natural subgroups of $QV$ that preserve (respectively) the linear and cyclic orderings of the ends of the infinite binary tree (and therefore bear a family resemblance to Thompson’s groups $F$ and $T$, respectively). The groups $\bar{Q}T$ and $\bar{Q}V$ are analogous groups that act as quasi-automorphisms on the union of an infinite binary tree with an isolated point. Nucinkis and St. John-Green show that the groups $QF$, $\bar{Q}T$, and $\bar{Q}V$ have type $F_{\infty}$, and also compute explicit finite presentations for these groups. Whether $QT$ and $QV$ are finitely presented (and thus, more particularly, of type $F_{\infty}$) is left as an open problem in [14].

The third author showed (in [11]) that $QV$ is a braided diagram group over a semigroup presentation. This description suggests an approach to proving the $F_{\infty}$ property for $QV$. Since [10] shows that a class of braided diagram groups (including Thompson’s group $V$) have type $F_{\infty}$, and in fact other classes of diagram groups were shown to have type $F_{\infty}$ in [8] and [10], it is at least plausible that some approach inspired by the theory of diagram groups could establish the $F_{\infty}$ property for $QV$ and $QT$. (We note that the original proofs that Thompson’s groups $F$, $T$, and $V$ have type $F_{\infty}$ were given by Brown [4] and by Brown and Geoghegan [5] in the 1980s.)

Nucinkis and St. John-Green show, however, that the hypotheses of the main theorem in [10] are satisfied by neither $QT$ nor $QV$. In fact, as also noted in [14], even the much more general main theorem of [15] does not apply to either of $QT$ and $QV$.

The goal of the present article is to extend the diagram-group methods of [8] and [10] to the groups $QF$, $QT$, $QV$, $\bar{Q}T$, and $\bar{Q}V$. We will show that all of these groups can be described using the theory of diagram groups over semigroup presentations. Indeed, all of these groups are diagram groups over the same semigroup presentation, namely $\mathcal{P}=\langle x,a\mid x=xax\rangle$, although the specific types of diagram vary from group to group. Three types of diagram groups have been considered in the literature: planar, annular, and braided diagram groups. All were introduced by Guba and Sapir in [12], which devotes by far the greatest attention to planar diagram groups (which are usually simply called diagram groups). The papers [9] and [10] consider the annular and braided diagram groups in more detail. Here we will introduce hybrid diagram groups that combine properties of multiple diagram group types. For instance, the group $QF$ is a special type of diagram group over $\mathcal{P}$, in which the diagrams exhibit both planar and braided behavior at the same time. We will use such hybrid diagrams to prove that the groups $QF$, $QT$, $QV$, $\bar{Q}T$, and $\bar{Q}V$ all act properly by isometries on CAT([math]) cubical complexes, and that all have type $F_{\infty}$. In fact, our methods extend with equal ease to the case of an arbitrary finite number of binary trees and isolated vertices (see Section 5), and the case of $n$-ary trees (for fixed $n\geq 2$) is different only in the details. It even seems likely that our argument generalizes to other, non-regular, trees, although this is more speculative, and we attempt no general statement about such cases here.

We note that it is probably possible to extend the main theorem of [15] to prove the $F_{\infty}$ property in the cases under consideration here. This is only a guess, however, since the authors claim little familiarity with the methods of [15].

We will now offer an outline of the argument. In Section 2, we will give a rapid introduction to the basic theory of diagram groups (including all three types: planar, annular, and braided), and describe the natural cubical complexes on which such groups act, including a description of the links of vertices. In Section 3, we will give careful definitions of the groups $QF$, $QT$, and $QV$, and describe how to represent elements of each group as “hybrid” diagrams. Section 3 also includes a description of natural complexes on which $QF$ and $QT$ act; these arise as convex (and thus CAT([math]) by [7]) subcomplexes of $QV$. (We will in fact confine our attention to $QF$, $QT$, and $QV$ alone, without sketching a general theory of “hybrid” diagrams. Nevertheless, we hope that the ideas indicated in Section 3 may be of some independent interest.) Section 4 shows that $QF$, $QT$, and $QV$ have type $F_{\infty}$. Our argument follows the long-established method given by Ken Brown in [4]. Section 5 sketches some possible further developments, including sketches of the proofs that $\bar{Q}T$ and $\bar{Q}V$ have type $F_{\infty}$ (as already proved by [14]), and the other generalizations briefly described above.

2. Braided diagram groups and actions on associated complexes

2.1. Definition of braided diagram groups

To define braided diagram groups, we must first define braided diagrams over semigroup presentations.

Definition 2.1**.**

(Braided diagrams over a semigroup presentation) Let $\mathcal{P}$ be a semigroup presentation; thus $\mathcal{P}=\langle\Sigma\mid\mathcal{R}\rangle$, where $\Sigma$ is a set (to be regarded as an alphabet) and $\mathcal{R}\subseteq\Sigma^{+}\times\Sigma^{+}$, where $\Sigma^{+}$ is the set of all non-empty positive words in the symbols $\Sigma$. We view the elements of $\mathcal{R}$ as equalities between elements of $\Sigma^{+}$ (i.e., as relations). For technical reasons, we impose the additional restriction that $(u,u)\notin\mathcal{R}$, for any $u\in\Sigma^{+}$.

A braided diagram $\Delta$ over $\mathcal{P}$ is a labelled ordered topological space formed by making identifications among three types of components: wires, transistors, and the frame.

• •

A wire is a homeomorphic copy of $[0,1]$. The “[math]” end is the bottom of the wire, and the “$1$” end is the top.

• •

A transistor is a homeomorphic copy of $[0,1]^{2}$. Each transistor has well-defined top, bottom, left, and right sides (in the obvious senses: the top is $[0,1]\times\{1\}$, the bottom is $[0,1]\times\{0\}$, etc.). These sides are part of the transistor’s definition. The top and bottom sides have equally obvious left-to-right orderings. (We make no use of any ordering of the sides.)

• •

The frame is a homeomorphic copy of $\partial[0,1]^{2}$. It has well-defined top, bottom, left, and right sides, just as a transistor does. The top and bottom sides have obvious left-to-right orderings.

To form a braided diagram $\Delta$ over $\mathcal{P}$, we begin with a finite non-empty collection $W(\Delta)$ of wires, a labelling function $\ell:W(\Delta)\rightarrow\Sigma$, a finite (possibly empty) collection $T(\Delta)$ of transistors, and a frame. Each endpoint of each wire is then attached either to a transistor or to the frame. The bottom of a wire is attached either to the top of a transistor or to the bottom of the frame; the top of a wire is attached either to the bottom of a transistor or to the top of the frame. Moreover, the images of any two wires in the quotient must be disjoint.

The resulting labelled oriented quotient space is called a braided diagram over $\mathcal{P}$ if the following two conditions are also satisfied:

1. (1)

Note that each transistor $T$ inherits a top and bottom label from the labels of the wires it touches. (The points at which the wires meet transistors are called contacts.) These labels are words in $\Sigma^{+}$, obtained by reading the labels of connecting wires from left to right.

Let $\ell_{top}(T)$ and $\ell_{bot}(T)$ denote the top and bottom labels, respectively. We require that $(\ell_{top}(T),\ell_{bot}(T))\in\mathcal{R}$ or that $(\ell_{bot}(T),\ell_{top}(T))\in\mathcal{R}$, for each transistor $T$. 2. (2)

For transistors $T_{1}$, $T_{2}$ of $\Delta$, write $T_{1}\preccurlyeq T_{2}$ if there is a wire whose bottom contact is a point on the top of $T_{1}$ and whose top contact is a point on the bottom of $T_{2}$. Let $<$ be the transitive closure of $\preccurlyeq$. We require that $<$ be a strict partial order on the transistors of $\Delta$.

(Equivalently: suppose that the braided diagram $\Delta$ is drawn in the plane, such that each transistor is enclosed by the frame, and the sides of the transistors and frame are parallel to the coordinate axes. We require that it be possible to arrange the transistors and the frame in this fashion in such a way that each wire can be embedded monotonically; i.e., so that the $y$-coordinate in the embedded image increases as we move from the bottom of the wire to the top.)

Definition 2.2**.**

(Planar and annular diagrams) Let $\Delta$ be a braided diagram over the semigroup presentation $\mathcal{P}$. If $\Delta$ admits an embedding $h:\Delta\rightarrow\mathbb{R}^{2}$ into the plane that preserves the left-right and top-bottom orientations on the transistors and frame, then we say that $\Delta$ is planar.

We say that $\Delta$ is annular if it can be similarly embedded in an annulus. Or, more precisely, suppose that we replace the frame $\partial([0,1]^{2})$ with a pair of disjoint circles, each of which is given the standard counterclockwise orientation, in place of the usual left-right orientations on the top and bottom of $\partial([0,1])^{2}$. We further give both circles basepoints, which are to be disjoint from all contacts. Transistors and wires are defined as before, and their attaching maps are subject to the same restrictions as before. We say that the resulting diagram is annular if the result may be embedded in the plane, again preserving the left-right orientations of the transistors. We think of the “top” circle as the inner ring of the annulus and the “bottom” circle as the outer ring.

[Here it may be helpful to view the transistors as having the counterclockwise orientation on their “top” and “bottom” faces, where the “top” faces the interior boundary circle of the annulus and the “bottom” faces the external boundary of the annulus.]

Definition 2.3**.**

(Equivalence of braided diagrams) Two braided diagrams $\Delta_{1}$ and $\Delta_{2}$ are equivalent if there is a homeomorphism $h$ between them, such that $h$ preserves the labelings of wires and all orientations (left-right and top-bottom) on all transistors and the frame.

Definition 2.4**.**

($(u,v)$-diagrams) Let $\Delta$ be a braided semigroup diagram; let $u,v\in\Sigma^{+}$. We can define the top and bottom labels of $\Delta$ by reading the labels of the wires that connect to the frame, from left to right, just as we defined the top and bottom labels of an individual transistor above. We say that $\Delta$ is a braided $(u,v)$-diagram if the top label of $\Delta$ is $u$ and the bottom label is $v$.

In some cases, it is not important to specify the bottom label. We say that $\Delta$ is a braided $(u,\ast)$-diagram if the top label of $\Delta$ is $u$, and the bottom label is arbitrary.

Definition 2.5**.**

(Concatenation) If $\Delta^{\prime}$ is a braided $(u,v)$-diagram and $\Delta^{\prime\prime}$ is a braided $(v,w)$-diagram, then the concatenation $\Delta^{\prime}\circ\Delta^{\prime\prime}$ is defined by stacking the diagrams, $\Delta^{\prime}$ on top of $\Delta^{\prime\prime}$.

Remark 2.6**.**

We note (for the sake of clarity) that the basepoints on the inner and outer circles of an annular diagram $\Delta$ are needed in Definitions 2.4 and 2.5. Here, the “top” label of $\Delta$ is to be read counterclockwise from the top (inner) basepoint, while the bottom label of $\Delta$ is similarly read counterclockwise from the outer circle’s basepoint.

If $\Delta_{1}$ and $\Delta_{2}$ are annular $(u,v)$- and $(v,w)$-diagrams (respectively, for $u$, $v$, $w\in\Sigma^{+}$), then the concatenation $\Delta_{1}\circ\Delta_{2}$ is the result of identifying the outer circle of $\Delta_{1}$ with the inner circle of $\Delta_{2}$ at the chosen basepoints (while also, of course, matching the other contacts in counterclockwise order).