#### by Arnaud Brothier

#### 1. A promenade from subfactors to CFT meeting Thompson’s group on the way

__\( F \)__and

__\( T. \)__

We will tell this story and its repercussions by first presenting subfactors, Thompson’s groups, CFT and explaining how they all became linked together. We will then introduce Jones’ technology for constructing actions of Thompson’s groups and will mainly focus on unitary representations. Finally, we will present how this latter framework led to a connection between Thompson’s groups and knot theory.

##### 1.1. Subfactors

*standard invariant*) that can be axiomatized — thanks to a reconstruction theorem due to Popa — and is described, for instance, by Jones’ planar algebra [1], [e10], [e13]. Structures like groups, subgroups and quantum groups can be encoded via subfactors, but also more exotic structures naturally appear in that context. Note that the Jones polynomial (the celebrated knot/link invariant) was defined using standard invariants of subfactors, creating a long bridge from operator algebras to low-dimensional topology [2]. Planar algebras are algebraic structures for which elements are composed in the plane rather than on a line. Compositions are encoded by planar diagrams that look like string diagrams used for monoidal categories. It is a collection of sets

__\( (P_n,\, n\geq 0) \)__(usually some finite-dimensional C*-algebras) that is a representation of the planar operad. Hence, any planar tangle like the following:

defines a map where, informally, one can place inside the inner discs
some elements of the __\( P_n \)__ (where __\( n \)__ must be the number of boundary
points) giving a new element of __\( P_m \)__ with __\( m \)__ the number of boundary
points of the outer disc. Hence, the last diagram gives a map from
__\( P_2\times P_3 \)__ to __\( P_3 \)__. Gluing tangles by placing one into an inner
disc of another provides an associative composition of maps and one
can modify tangles by isotopy without changing the associated map.

#### 1.2. Thompson’s groups

__\( F\subset T\subset V \)__, sometimes called chameleon groups for good reason, where

__\( F \)__is the group of piecewise linear homeomorphisms of the unit interval with slopes powers of 2 and finitely many breakpoints at dyadic intervals [e11]. Elements of

__\( F \)__map one (standard dyadic) partition into another in an order-preserving way, being affine on each subinterval. Larger groups

__\( T,V \)__are defined similarly but their elements are allowed to permute subintervals of the associated partitions in a cyclic way, or in any possible way, respectively. In particular,

__\( T \)__still acts by homeomorphisms but on the circle rather than on the interval. Those groups have been extensively studied as they naturally appeared in various fields of mathematics such as infinite group theory, homotopy and dynamical systems, and follow very unusual behavior [e5], [e12]. A famous open problem is to decide whether

__\( F \)__is amenable or not, but even more elementary questions are still open, such as whether

__\( F \)__is exact or weakly amenable in the sense of Cowling and Haagerup [e7]. It is surprising to meet those discrete groups while considering very continuous structures like CFT and subfactors, but we will see that

__\( T \)__appears as a discretization of the conformal group. Moreover, elements of

__\( T \)__can be described by diagrams of trees, suggesting a connection with Jones’ planar algebras and thus with subfactors.

##### 1.3. From CFT to subfactors and back

For us, a conformal net or a CFT is the collection of field algebras
localized on intervals of the circle (spacetime regions), on which the
diffeomorphism group acts, and that is subject to various axioms
coming from physics
[e9].
Representation theory of a conformal net looks like very much the algebraic data of a subfactor and one wants to know how similar they are.
From a conformal net one can reconstruct a subfactor.
However, the converse is fairly mysterious and only specific examples have been worked out, missing the most fascinating ones: the exotic subfactors (subfactors not coming from quantum groups).
It is a fundamental question whether such a reconstruction always
exists (“Does every subfactor have something to do with a CFT?”) and
Jones has been trying very hard to answer it
[3],
[e21],
[e23].
One of his attempts started as follows
[4]:
given a subfactor we consider its planar algebra __\( P=(P_n,\, n\geq 0) \)__.
The idea is then to interpret the outer boundary of a planar tangle as
the spacetime circle of a CFT. Given any finite subset __\( X \)__ of the
dyadic rationals of the unit disc we consider __\( P_X \)__, a copy of
__\( P_{|X|} \)__, where all boundary points on the outer disc of planar
tangles are in __\( X \)__. This __\( X \)__ provides a partition of the unit disc. We
want to be able to refine this partition __\( X \)__ into a thinner one __\( Y \)__ by
adding middle points and to embed __\( P_X \)__ inside __\( P_Y \)__ (giving us a
directed system). This is done using a fixed element __\( R\in P_4 \)__ that
we think of as a trident-like
diagram:

Here is one example that explains how we build a map __\( P_X\to P_Y \)__.
Consider a finite subset of
points
__\[
\textstyle X:=\bigl\{0,\frac18,\frac14,\frac12,\frac34\bigr\}
\]__
of the
circle identified with the torus __\( \mathbb{R}/\mathbb{Z} \)__.
Placing those points on the disc we obtain a partition with intervals
__\[
\textstyle
\bigl(0,\frac18\bigr),\ \bigl(\frac18,\frac14\bigr),\ \ldots,\ \bigl(\frac34,1\bigr).
\]__
Let us refine this partition by
splitting the two consecutive intervals __\( \bigl(0,\frac18\bigr) \)__ and __\( \bigl(\frac18,\frac14\bigr) \)__ in
two equal halves. This refined partition is characterized by the larger
subset of points
__\[
\textstyle
Y:=X\cup \bigl\{\frac1{16},\frac3{16}\bigr\}
\]__
in which we added the middle
points of __\( \bigl(0,\frac18\bigr) \)__ and __\( \bigl(\frac18,\frac14\bigr). \)__ Consider a planar tangle with one
inner disc. Place the points of __\( X \)__ on the inner disc and the points
of __\( Y \)__ on the outer disc. For common elements of __\( X \)__ and __\( Y \)__ we draw a
straight line from the inner to the outer disc. In order to connect
the two new points of __\( Y \)__ we use our trident-like diagram.
We obtain the following tangle:

By definition of the planar operad this tangle encodes a map from
__\( P_X \)__ to __\( P_Y \)__ and under a certain condition on __\( R \)__ this latter map is
injective. Continuing this process of refinement of finite partitions
we obtain at the limit the dense subset of dyadic rationals of the
circle and obtain an obvious notion of support defining localized
field algebras exactly like in (physics) lattice theory. Moreover, we
can rotate and perform some local scale transformations but only using
those behaving well with dyadic rationals. *This group of
transformation is none other than Thompson’s group *__\( T. \)__ Moreover,
the tree-diagram description of elements of __\( T \)__ can be explicitly used
to understand this action simply by sending a branching of a tree to a
trident __\( R \)__ in the planar algebra.
We obtain some kind of discrete CFT with __\( T \)__ replacing the diffeomorphism group and field algebras localized on intervals of the circle.
At this point, the hope was to perform a continuum limit and obtain an
honest CFT but unfortunately strong discontinuities arise and the CFT
goal was out of reach
[5];
see also
[e29].

The story could have stopped here but in fact this failed attempt opened whole new fields of research in both mathematics and physics. Indeed, accepting that the continuum limit cannot be done provides physical models relevant at a quantum phase transition with Thompson’s group for symmetry [6], [e28]. Moreover, Jones’ construction paired with models in quantum loop gravity leads to lattice-gauge theories, again with Thompson’s group symmetry [e30], [e27]. The physics described by Jones mathematical model is rather discontinuous and predicts different phenomena than CFT. Jones suggested the following laboratory experiment which would confront the two theories: set up a quantum spin chain and observe the correlation number associated to small translations. Approach a quantum phase transition. According to CFT the correlation number stays close to 1 but Jones’ model with Thompson group for symmetry predicts that this number becomes small. On the mathematical side, Jones discovered a beautiful connection between knot theory and Thompson’s groups by using the planar algebra of Conway tangles [9]. Moreover, he provided a whole new formalism for constructing unitary representations and evaluating matrix coefficients for Thompson’s groups that generalizes the planar algebraic construction [5].

#### 2. Actions and coefficients

After presenting how Thompson’s groups were found in between subfactors and CFT we now present the general theory for constructing groups and actions from categories and functors that we illustrate with Thompson’s groups. Note that this formalism was not developed for the sake of generality but rather to understand better Thompson’s group and other related structures. Jones’ research is driven by the study of concrete and fundamental objects in mathematics such as Temperley–Lieb–Jones algebras, Haagerup’s subfactor, Thompson’s groups, braid groups, etc. His approach is to use or create whatever formalism is pertinent for better understanding those objects, leading to brand new theories like subfactor theory, planar algebras and today Jones actions for groups of fractions. We follow Jones’ attitude by presenting a general formalism but always accompanied by key examples and applications.

##### 2.1. Groups of fractions

__\( \mathcal{F} \)__of finite ordered rooted binary forests where the objects are the natural numbers and morphisms

__\( \mathcal{F}(n,m) \)__, the set of forests with

__\( n \)__roots and

__\( m \)__leaves that we consider as diagrams with roots on the bottom and leaves on top. Composition is obtained by vertical concatenation. For example,

__\( F \)__is described by an equivalence class

__\( t/s \)__of pairs of

*trees*

__\( (t,s) \)__having the same number of leaves, where the class

__\( (t,s) \)__is unchanged if we add a common forest on top of each tree [e6], [e11]. This comes from the identification between finite binary rooted trees and standard dyadic partitions of the unit interval. We often described this pair with two trees:

__\( s \)__on the bottom and

__\( t \)__reversed on top. For example, if

__\[ \frac{t}{s}\cdot \frac{s}{r} = \frac{t}{r} \]__and thus

__\[ \biggl(\frac{t}{s}\biggr)^{\!-1} = \frac{s}{t}. \]__This corresponds to formally inverting trees and considering morphisms from 1 to 1 inside the universal groupoid of the category of forests

__\( \mathcal{F} \)__, where the group

__\( F \)__is identified with the automorphism group of the object 1 inside this latter groupoid. Groups arising in this way are called

*groups of fractions*. Considering trees together with cyclic permutations (affine trees) or all permutations (symmetric trees) we obtain the larger Thompson’s groups

__\( T \)__and

__\( V \)__and if we consider forests with

__\( r \)__roots instead of trees we get Higman–Thompson’s groups. Taking braids, we obtain the braid groups, and taking a topological space as a collection of objects with paths (up to homotopy) for morphisms we get the Poincaré group. All of this was observed long ago in a categorical language in [e1] and for the particular example of Thompson’s groups [e6] that was rediscovered in different terms by Jones.

##### 2.2. Jones actions

Jones found a machine to produce in a very explicit manner *actions* of groups of fractions.
Given a functor __\( \Phi:\mathcal{F}\to\mathcal{D} \)__ he constructed an action __\( \pi:F\curvearrowright X \)__ that we call a *Jones action*.
Formally, for a covariant functor and a target category with sets for objects, the space __\( X \)__ is the set of *fractions* __\( t/x \)__ that are classes of pairs __\( (t,x) \)__ with __\( t \)__ a tree, __\( x\in\Phi(n) \)__ with __\( n \)__ being the number of leaves of __\( t \)__ and where the equivalence relation is generated
by
__\[
(t,x)\sim (ft,\Phi(f)x)
\]__
for any forest __\( f \)__.
The Jones action is then defined
as
__\[
\pi\biggl(\frac{s}{t}\biggr) \frac{t}{x} = \frac{s}{x}.
\]__
We sometimes want to complete this space
with respect to a given metric, and this is what we do if __\( \mathcal{D} \)__ is the category of Hilbert spaces.
Observe that __\( X \)__ is defined in the same way as the group of fractions except that now the denominator is in the target category and the equivalence relation is defined using the functor __\( \Phi. \)__
Making __\( \mathcal{F} \)__ monoidal by declaring that the tensor product of forests is the horizontal concatenation, we obtain that __\( \mathcal{F} \)__ is generated by the single morphism __\( Y, \)__ i.e., the tree with two leaves.
Hence, (monoidal) functors __\( \Phi:\mathcal{F}\to\mathcal{D} \)__ correspond to
morphisms
__\[
R:=\Phi(Y)\in\operatorname{Hom}_\mathcal{D}(a, a\otimes a)
\]__
in the target category __\( \mathcal{D} \)__.
In particular, a Hilbert space __\( \mathfrak{H} \)__ and
an *isometry*
__\[
R:\mathfrak{H}\to\mathfrak{H}\otimes\mathfrak{H}
\]__
provide a unitary representation of Thompson’s group that we call a *Jones representation*.
Using string diagrams to represent morphisms in a monoidal category we can interpret a functor __\( \Phi:\mathcal{F}\to\mathcal{D} \)__ as taking the diagram of a forest and associating the exact same diagram but in the different environment of the target category __\( \mathcal{D} \)__.
This procedure is nothing other than replacing each branching in a
forest by an instance of the morphism __\( R \)__:

Note that for technical reasons we might use morphisms with four boundary points rather than three (if, for instance, one wants to work with subfactor planar algebras or Conway tangles that only have even numbers of boundary points) and thus consider a map of the form

We give credit to Jones for those actions, even if some of the ideas were already around; however, the construction with a direct limit was completely new. We are grateful to Matt Brin for a very nice explanation of the state of the art before Jones’ work. “What was known was that certain automorphism groups contained Thompson’s groups. How they acted was never under investigation and the fact that the actions could be manipulated to get desired properties never even occurred to anyone.”

##### 2.2.1. Planar algebraic examples

__\( P \)__with a one-dimensional 0-box space (diagrams without boundary points correspond to numbers) and choose an object

__\( P \)__is equipped with an inner product which consists of connecting two elements of

__\( P_n \)__via

__\( n\text{ strings} \)__like

Then __\( R \)__ defines a unitary representation and moreover has a favorite
vector called the vacuum vector corresponding to a straight line in
the planar algebra. The positive definite function associated to the
vacuum vector __\( \Omega \)__ is then a closed diagram inside __\( P \)__ which is
equal to the following if we consider the group element __\( t/s \)__
of (2.1):

__\( P \)__where it corresponds to a number (via the

*partition function*). This number can be explicitly computed using

*skein relations*(diagrammatic rules for reducing diagrams, such as (2.2), analogous to relations for a presented group) of the planar algebra chosen. Jones called them

*wysiwyg representations*(“what you see is what you get”) [8]. Planar algebras have been extensively studied in the past two decades and we know today many interesting examples with fully understood skein relations providing candidates for wysiwyg representations. Using a certain class of planar algebras, (trivalent categories studied by Peters, Morrison and Snyder [e20]), Jones constructed an uncountable family of

*mutually inequivalent*wysiwyg unitary representations that are all

*irreducible*.

Those latter examples emanate from the planar algebraic approximation of CFT and keep some geometric flavor. Next we present examples that somehow forget the geometric structure of planar algebras but can be defined in a very elementary way.

##### 2.2.2. Analytic examples

__\( \operatorname{Hilb} \)__with

*isometries*for morphisms. There are various monoidal structures

__\( \odot \)__we can equip

__\( \operatorname{Hilb} \)__with such as the classical tensor product or the direct sum. Free products can also be done but one has to consider a slightly different category where objects are pointed Hilbert spaces

__\( (H,\xi) \)__, where

__\( \xi \)__is a unit vector and morphisms are isometries sending the chosen unit vector to the other one; see [e8]. Each case provides Jones representations by taking an isometry

__\[ R:\mathfrak{H}\to\mathfrak{H}\odot\mathfrak{H} \]__for the chosen monoidal structure

__\( \odot \)__. The second case can be written as

__\( R=A\oplus B \)__with

__\( A,B:\mathfrak{H}\to\mathfrak{H} \)__satisfying the

*Pythagorean*identity

__\begin{equation} \tag{2.4} \label{eq:Pyth}A^*A+B^*B=1. \end{equation}__We call the

*Pythagorean algebra*the universal C*-algebra generated by this relation and observe that a representation of this latter algebra provides a Jones representation of Thompson’s groups [10]. Moreover, it has interesting quotient algebras such as the Cuntz algebra, noncommutative tori, and Connes–Landi spheres [e2], [e3], [e4], [e14]. Taking any unit vector

__\( \xi\in \mathfrak{H} \)__we obtain a positive definite function as matrix coefficient. It can be computed as follows. Consider an element of

__\( F \)__written

__\( t/s \)__. Place

__\( \xi \)__at the root of

__\( s \)__and make it go to the top by applying

__\( A\oplus B \)__at each branching. We obtain on top of each leaf a word in

__\( A,B \)__applied to

__\( \xi \)__. We do the same thing for

__\( t \)__and then take the sum of the inner product at each leaf. For example, if

__\( t/s \)__is the example of (2.1) we obtain the inner product

__\[ \langle A\xi \oplus AB\xi \oplus BB\xi , AA\xi\oplus BA \xi \oplus B\xi\rangle \]__and the procedure in making

__\( \xi \)__going from the bottom to the top of the tree

__\( s \)__is described by the following diagram:

The formula of this coefficient for elements of the larger group __\( T \)__
is similar up to permuting cyclically the order of the vectors in the
direct sum and can be extended to __\( V \)__ by considering *any*
permutations. Many interesting representations and coefficients of
Thompson’s groups can be created in that way. If __\( A=B \)__ are real
numbers equal to __\( 1/\sqrt 2 \)__, then we recover the Koopman
representation __\( T\curvearrowright L^2(\mathbb S^1) \)__ induced by the usual action of
__\( T \)__ on the circle. In more
detail: fix a tree __\( s \)__ and a
complex number __\( \xi \)__. Following the procedure explained by the
diagram above we obtain that each leaf __\( \ell \)__ of __\( s \)__ is decorated by
__\( 2^{-d^\ell_s/2} \xi \)__, where __\( d_s^\ell \)__ is the distance from the leaf
to the root. This latter number corresponds to __\( \xi \)__ times the
square root of the length of the interval __\( I_s^\ell \)__ associated to
the leaf. Taking a second tree __\( t \)__ such that __\( g=t/s\in F \)__ we
obtain that the contribution of the inner product associated to
__\( \xi=1 \)__ and __\( g \)__ at a leaf __\( \ell \)__ is __\( 2^{(d^\ell_s-d^\ell_t)/2} \)__ that
is the square root of the slope of __\( g \)__ when restricted to
__\( I_s^\ell \)__. From this observation it is not hard to conclude.
Thanks to the flexibility of Jones’ formalism, we can easily deform
this representation by replacing __\( 1/\sqrt 2 \)__ by two different real or
complex numbers __\( v \)__ and __\( w \)__ with __\( |v|^2+|w|^2=1 \)__ obtaining various
paths between the Koopman and the trivial representations where
the former appears when __\( v \)__ or __\( w \)__ is equal to zero. Using the free
group we obtain the
map
__\[
g\in F\mapsto \operatorname{Measure}(x\in (0,1) : gx=x)
\]__
as
a diagonal matrix coefficient, and it is then positive definite. Other
examples arise by taking representations of quotients of the
Pythagorean algebras, providing interesting
families of representations.
One can also
use this approach for constructing representations of
such quotient algebras: with the help of
Anna Marie Bohman
and
Ruy Exel,
Jones and I could relate precisely representations of the Cuntz
and the Pythagorean algebras, obtaining new methods for practical
constructions of representations of the former.

If we choose the monoidal structure to be the classical tensor product
of Hilbert spaces, then any isometry
__\[
R:\mathfrak{H}\to\mathfrak{H}\otimes\mathfrak{H}
\]__
provides a
unitary representation of __\( V \)__. Matrix coefficients associated to
__\( \xi,\eta\in\mathfrak{H} \)__,
__\[
\biggl\langle \pi\biggl(\frac{t}{s}\biggr) \xi,\eta\biggr\rangle,
\]__
can be
computed as above but where we need to perform an inner product of two
vectors in a tensor power of __\( \mathfrak{H} \)__ instead of a direct sum, where each
tensor power factor corresponds to a leaf of the tree __\( t \)__. Interesting
and manageable examples arise when __\( R\xi \)__ is a finite sum of
elementary tensors and thus matrix coefficients are then computed in
an algorithmic way. Here is one story concerning those representations
and how they can be manipulated and used.

During February 2018 Jones and I met one week in the beautiful coastal
town of Raglan in New Zealand to finish up the paper on Pythagorean
representations and to enjoy the kite-surf spot a bit. During
this stay Jones told me that the absence of Kazhdan property (T) for
__\( F,T,V \)__ could be trivially proved via his recent formalism. Indeed,
this can be done using maps like
__\[
R\xi=u\xi\otimes\zeta ,
\]__
where __\( \zeta \)__
is a fixed unit vector and __\( u \)__ an isometry. For example, this map
and the pair of trees __\( t/s \)__ of (2.1)
give the
following matrix coefficient:
__\[
\biggl\langle \pi\biggl(\frac{t}{s}\biggr) \zeta , \zeta \biggr\rangle =
\langle u\zeta\otimes u\zeta\otimes \zeta , u^2\zeta\otimes \zeta\otimes
\zeta\rangle = |\langle \zeta,u\zeta\rangle|^2.
\]__
By considering a family of those pairs __\( (u,\zeta) \)__ and making __\( \langle
u\zeta,\zeta\rangle \)__ tend to
1, we obtain an almost invariant vector
but no invariant one in the associated Jones representation.

Moreover, he showed me how to create the left regular representation of __\( F \)__ via a tensor product construction where __\( \mathfrak{H}=\ell^2(\mathbb N) \)__ and
__\[
R\delta_n=\delta_{n+1}\otimes\delta_{n+1}=\delta_{n+1,n+1}.
\]__
Indeed, if __\( t \)__ is a tree, then using the functor __\( \Phi \)__ we get __\( \Phi(t)\delta_0 = \delta_{w_t}, \)__ where __\( w_t \)__ is the list of distances between each leaf of __\( t \)__ to its root. Since this characterizes the tree __\( t \)__ we obtain that the cyclic component of the Jones representation associated to the vector __\( \delta_0 \)__ is the left regular representation of __\( F \)__.

Those two facts made me very excited. Showing that Thompson’s groups
are not Kazhdan groups is a difficult result that stayed open for
quite some time. Jones’ proof being so effortless gave hope to obtain
stronger results with more
elaborate techniques. The regular
representation has coefficients vanishing at infinity and thus one
might be able to construct
others of
that kind.
For this purpose we
started to think about deforming the isometry __\( R\delta_n=\delta_{n+1,n+1} \)__,
obtaining paths between the trivial and the left regular
representations and new coefficients. Going back home during the very
long journey from Raglan to Rome I only thought about those
deformations. When I landed I had more or less a full proof showing
that __\( T \)__ has the Haagerup property improving the absence of
the Kazhdan
property but only for the intermediate Thompson’s group __\( T \)__. I wrote
to Jones about it and we decided to write a short paper giving the two
proofs: __\( F,T,V \)__ are not Kazhdan groups and __\( T \)__ has the Haagerup
property
[11].
Even though those results are not
optimal and already known (Farley showed that __\( V \)__ has the Haagerup
property
[e16])
they display the power of Jones’ new
techniques. One intriguing fact is that the maps constructed by
Farley (the one associated to his cocycle) coincide with ours on
Thompson’s group __\( T \)__ but differ on the larger group __\( V \)__ and it is
still unclear how to build them using Jones representations. Another
interesting problem would be to construct Farley actions on CAT(0)
cubical complexes via Jones actions using the appropriate target
category
[e15].

A year later, new results were proved regarding analytical properties
of groups. Choose a group __\( \Gamma \)__ and a single group morphism
__\[
g\in\Gamma\mapsto (a_g,b_g)\in\Gamma\oplus\Gamma.
\]__
This provides a (monoidal)
functor from the category of forests to the category of groups and
thus a Jones action of __\( V \)__ on a limit group. One can then consider the
semidirect product. Choosing the trivial embedding __\( g\in\Gamma\mapsto
(g,e) \)__ we obtain the (permutational and restricted) wreath product
__\( \bigoplus_{\mathbb{Q}_2} \Gamma\rtimes V \)__, where __\( V \)__ shifts the indices
via the usual action __\( V\curvearrowright \mathbb{Q}_2 \)__, where
__\( \mathbb{Q}_2 \)__ is the set of dyadic rationals on the unit circle.
Now comes a trivial but *key observation*: this new group can
be written as a group of fractions where the new category is basically
made of forests but with leaves labeled by elements of __\( \Gamma. \)__
Composition of forests with group elements in this latter category,
constructed with the map __\( g\mapsto (a_g,b_g) \)__, is expressed by the
equality

Note that a similar construction was observed by Brin using
Zappa–Szép products,
which he used to define the braided Thompson’s
group
[e18];
see also
[e17].
Since
it is a group of fractions, we can then apply Jones’ technology for
constructing representations and coefficients of this larger group.
Using this strategy I was able to show that those wreath products have
the Haagerup property when __\( \Gamma \)__ has it, which was out of reach by
other known approaches
[e26].

##### 2.3. Connection with knot theory

Knot theory and Thompson’s groups are connected using the technology presented above. This has been very well explained in a recent expository article of Jones so I will be brief [9]. The connection comes from the idea to consider functors from forests to the category of Conway tangles that are roughly speaking strings inside a box possibly attached to the top and/or the bottom that can cross like

We ask that those diagrams be invariant under isotopy and the three Reidemeister moves. We define a functor from (binary) forests to Conway tangles by replacing each branching by one crossing as follows:

Consider
an element of __\( g=t/s\in F \)__ with __\( t,s \)__ trees,
where we put
__\( s \)__ on the bottom, __\( t \)__ upside down on top and connect their roots. We
then apply our transformation that
replaces each branching by a
crossing obtaining a link.

The procedure is the following for the example of (2.1):

*Jones–Thompson index*of a link (what is the minimal number of leaves necessary for a couple of trees to form this link). This latter index for links can be defined for

__\( k \)__-ary trees,

__\( k\geq 2 \)__, instead of binary trees, giving a discrete-parameter family of invariants for links.

The connection with links provided a new point of view on Thompson’s
group elements. One can then ask whether the link associated to an
element of __\( F \)__ is orientable or not. It turns out that the set of all
__\( g\in F \)__ giving an orientable link forms a subgroup __\( \vec F\subset F \)__
known today as the *Jones subgroup*. This subgroup can be
defined in a number of ways using diagram groups, skein theory,
stabilizers and is even equal to a group of fractions
[4],
[e19],
[e22],
[7].
Golan
and
Sapir
were able to prove the striking result that __\( \vec F \)__
is isomorphic to Thompson’s group __\( F_3 \)__ associated to 3-adic numbers
and moreover is equal to its commensurator, implying that the
associated quasiregular representation is irreducible
[e19].
The different definitions of __\( \vec F \)__ suggested
various natural generalizations of it: a circular Jones subgroup __\( \vec
T\subset T \)__; from the diagram group approach Golan and Sapir obtained
an increasing chain of subgroups __\( \vec F_n\subset F, \)__ __\( n\geq 2; \)__ and the
stabilizer definition provided an uncountable family __\( F_{(I)}\subset
F \)__ parametrized by the famous Jones’ range of indices for subfactors
__\[
\biggl\{4\cos\biggl(\frac{\pi}n\biggr)^{\!2}: n\geq 4\biggr\}\cup [4,\infty)
\]__
[1].
Using skein theory of planar
algebras
Ren
interpreted differently __\( \vec F \)__ reproving that it is
isomorphic to __\( F_3 \)__ and interestingly,
he and
Nikkel
showed that
__\( \vec T \)__ is *not* isomorphic
to __\( T_3 \)__ but is similar from a
diagram-group perspective
[e22],
[e24].
Golan and
Sapir proved properties for the subgroups __\( \vec F_n\subset F \)__
similar to those for __\( \vec F\subset F \)__, obtaining an infinite family of irreducible
representations and isomorphisms __\( \vec F_n \simeq F_{n+1} \)__ (Thompson’s
group associated to __\( (n+1) \)__-adic numbers)
[e19].
The
uncountable family of __\( F_{(I)} \)__ is less exciting as it generically
provides trivial subgroups, except of course for __\( \vec F \)__ which
corresponds to the first nontrivial Jones’ index 2
[e25].
The Jones subgroup __\( \vec F \)__ and the study around it summarize well the
interplay of various fields in this brand new framework of Jones and
how ideas, say from knot theory or skein theory, can be then applied
in group theory and vice versa.

#### 3. Conclusion

The recent technology of Jones regarding Thompson’s groups has provided new perspectives and connections for and between groups of fractions, knot theory, subfactor theory and quantum field theory. This complements previous beautiful connections that Jones made more than 35 years ago with his celebrated polynomial. This is only the very beginning of this development and various exciting research directions remain untouched. There have been already beautiful applications and promising techniques developed which augur a bright future.