 # Celebratio Mathematica

## Vaughan F. R. Jones

### On Jones’ connections between subfactors, conformal field theory, Thompson’s groups and knots

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

From the very be­gin­ning the work of Jones has been mo­tiv­ated by and con­nec­ted to math­em­at­ic­al phys­ics. His the­ory of sub­factors is linked to quantum field the­ory and in par­tic­u­lar to chir­al con­form­al field the­ory (CFT), which has been form­al­ized in vari­ous ways, such as ver­tex op­er­at­or al­geb­ras or con­form­al nets. Those lat­ter math­em­at­ic­al ob­jects give sub­factors, and some sub­factors provide con­form­al nets. It is by try­ing to find a sys­tem­at­ic re­con­struc­tion that Jones un­ex­pec­tedly met Richard Thompson’s groups $$F$$ 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
Sub­factors are in­clu­sions of von Neu­mann al­geb­ras with trivi­al cen­ter, which are called factors. They carry a rich al­geb­ra­ic struc­ture (the stand­ard in­vari­ant) that can be ax­io­mat­ized — thanks to a re­con­struc­tion the­or­em due to Popa — and is de­scribed, for in­stance, by Jones’ planar al­gebra , [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 . 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

Richard Thompson defined three 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 , [e21], [e23]. One of his at­tempts star­ted as fol­lows : 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 ; 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 , [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 . 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 .

#### 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
The gen­er­al idea is that a cat­egory gives a group and a func­tor an ac­tion. Our lead­ing ex­ample is the cat­egory $$\mathcal{F}$$ 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,
It has been ob­served that an ele­ment of Thompson’s group $$F$$ 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
The group struc­ture is giv­en by the for­mula $\frac{t}{s}\cdot \frac{s}{r} = \frac{t}{r}$ 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
Let us com­pute some coef­fi­cients with this tech­nique. Start with a planar al­gebra $$P$$ with a one-di­men­sion­al 0-box space (dia­grams without bound­ary points cor­res­pond to num­bers) and choose an ob­ject
sat­is­fy­ing
As­sume that $$P$$ 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):

but viewed in­side $$P$$ 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”) . 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
Let us con­sider the whole cat­egory of Hil­bert spaces $$\operatorname{Hilb}$$ 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 . 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 . 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 . 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):

Smooth­ing out this dia­gram we ob­tain the fol­low­ing knot:
Jones proved the the­or­em that every link can be ob­tained in that way, con­clud­ing that “Thompson’s group is as good as the braid groups for pro­du­cing links”. This opened a com­pletely new land of study in which Jones already sug­ges­ted nine ex­pli­cit re­search prob­lems which ex­plore the group struc­ture ana­logy between Thompson’s and braid groups (, Sec­tion 7). Ex­amples of prob­lems are find­ing a Markov the­or­em for Thompson (what are the re­la­tions between two Thompson’s group ele­ments that give the same link) or de­cid­ing the 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 , [e19], [e22], . 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 $\biggl\{4\cos\biggl(\frac{\pi}n\biggr)^{\!2}: n\geq 4\biggr\}\cup [4,\infty)$ . 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.

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