## Abstract

A machinery developed by the second author produces a rich family of unitary representations of the Thompson groups *F*, *T* and *V*.
We use it to give direct proofs of two previously known results.
First, we exhibit a unitary representation of *V* that has an almost invariant vector but no nonzero *F* and *V* does not have Kazhdan’s property (T) (though Reznikoff proved it for subgroups of *T*).
Second, we construct a one parameter family interpolating between the trivial and the left regular representations of *V*.
We exhibit a net of coefficients for those representations which vanish at infinity on *T* and converge to 1 thus reproving that *T* has the Haagerup property after Farley who further proved that *V* has this property.

## Introduction

Let *n* trees all of which are trivial except the *i*th one that has two leaves.
Since any tree is a composition of *F* that we extend to *V* via permutations of the tensors; see Section 1 for more details.

In Section 2, we construct a unitary representation of *V* having an almost invariant vector but no nonzero *V* does not have Kazhdan’s property (T) [8].
Recall that *T* was proved to not have property (T) initially by Reznikoff [11].
Another proof consists in embedding *T* in the group of diffeomorphisms of the circle via the smoothing methods of Ghys and Sergiescu [6] or Thurston (described in [2]) and using results of Navas concerning diffeomorphism groups that imply that *T* does not have property (T) [10].

In Section 3, we give a one parameter family (*V* interpolating the trivial and the left regular one.
We define some positive definite maps *T*.
They actually coincide on *T* with the family of maps constructed from the proper cocycle of Farley (see Remark 1), but differ on the larger group *V*.
Hence they provide a *T*, thus proving Farley’s result that *T* has the Haagerup property (though Farley actually proved it for all of *V*) [1, 4].
Note that a geometric approach to Haagerup and Kazhdan properties for Thompson’s groups follows from the actions on CAT(0) cube complexes constructed by Farley [3, 5] (which is related to Chatterji, Drutu and Haglund’s characterization of these properties via median metric spaces).

## 1 Definitions and notations

We briefly recall the construction of actions of groups of fractions for the particular cases of *F*, *T* and *V* and refer to [7, 2] for more details.
Let *n* roots and *m* leaves.
We think of them as planar diagrams in the plane *q* the forest *p*, where the *i*th root of *p* is attached to the *i*th leaf of *q*.
We obtain a diagram in the strip *n* roots and *i*th tree of

Consider the set of pairs of *trees**p*.
We write *F*.

Now consider the category of symmetric forests *m* elements.
Graphically, we interpret a morphism *p* in the strip *m* segments *m* distinct points in *p* such that *i*th leaf of *p*.
Given symmetric forests *i*th tree of *p*.
Then we define the composition of morphisms by
*p* by permuting its trees such that the *i*th tree of *p* and *i*th segment *V* is isomorphic to the group of fractions of the category *V* is an equivalence class of a pair of symmetric *trees*.
Consider *s* and *t*, respectively.
The element *g* acting on

Consider the cyclic group *affine* forests, where *T*.
We will often identify

We say that a pair of symmetric trees *t* and such that

Let

for

where the equivalence relation

This quotient space has a pre-Hilbert structure given by *t* such that the embedding *V* as follows:

Note that if

Consider an orthonormal basis *R* can be thought of as a possibly infinite matrix with three indices *f* as a partition function.
Given a forest *f* with *n* roots and *m* leaves, we define the set of states *f* as maps ω from the edges of *f* to the set of indices *I*.
A *vertex* of *f* is a trivalent vertex, and thus roots and leaves are not vertices.
If ω is a state on *f* and *v* a vertex, then we put *v* and *v* which goes to the left and right, respectively.
Consider some multi-indices *k*th root of *f* and *f*.

## Lemma 1.

*Using the notations of above, we have*

*with the convention that a product (resp. a sum) over an empty set is equal to one (resp. zero).*

The infinite sum converges since the scalars *f* has only one vertex (roots and leaves are not considered as vertices), and so there is only one term in the product that is

that we represent with the spin of an edge (i.e., the image by ω of this edge) next to it, then

## 2 Kazhdan’s property (T)

Recall that a countable discrete group *G* has Kazhdan’s property (T) if any unitary representation having an almost invariant vector has in fact a nonzero invariant vector [8].

If

is an isometry which provides us a unitary representation

and let

Define the element

where *a*.
Similarly, define

and observe that

and thus

If

By linearity and density, we obtain that *any* vectors *n* and a unit vector

for *n* as large as we want, which implies a contradiction since

Consider *V*, and put *m*th slot and zero elsewhere.
We claim that the sequence *V*-invariant vector.
Fix *m* big enough, there exists *m*, we can also assume that there exists

Since *p* (resp. *q*) has length smaller than or equal to *m*).
This implies that any component of

Therefore,

and thus *V*-invariant vector.
Since π does not have any nonzero *V* does not have property (T).

## 3 Haagerup property

Recall that a countable discrete group *G* has the Haagerup property if there exists a sequence *G* and such that

Consider the free group

This defines a functor

is particularly simple since

and zero elsewhere.

Consider the case when *f* be a forest with *n* roots and *m* leaves, and observe that

The *i*th component *f* to its *i*th leaf such that a left turn (resp. right turn) contributes in adding the letter “*a*” (resp. the letter “*b*”).
For example, if

The next lemma proves that if *t* is a tree, then the set of words in the tuple *t*.

## Lemma 2.

*Consider two trees *

## Proof.

We prove the lemma by induction on the number of leaves

Suppose the result is true for any *k* between 1 and *n*, and consider *a* (resp. the letter *b*) if and only if *i* is a leaf of *t*, and thus necessarily σ realizes a bijection from the leaves of

The lemma implies that *V*.
Indeed, consider some symmetric trees

This is nonzero (and then equal to one) if and only if *V*.
If *t* is a tree with *n* leaves, then *n* tensors, and thus coefficient (3.1) is always equal to one for any choice of *V* provides an interpolation between the trivial and the left regular representations.

Consider the family of positive definite maps

We will show that they vanish at infinity for *R* instead of *t* with *n* leaves, we have

for any multi-index *t* such that the spin (i.e., the value of ω) at each of its edges is the trivial group element *v* of the tree *w* of *u* of

where *t* (i.e., the number of nontrivial trees of *n* leaves has *t* is the word in *n*-tuple, and note that, by definition, the multi-index *z* of *t* provides a unique state

We obtain

where *t*.

Consider a pair of symmetric trees

Fix a *reduced* pair of *affine* trees *T* since our trees are affine.
We will show that all the terms in the sum (3.2) are equal to zero but one.

## Lemma 3.

*Consider some forests *

## Proof.

Observe that if *f* is a forest, then the word *a* (resp. a power of *b*) if and only if *i* corresponds to the first leaf (resp. the last leaf) of a tree of *f*.
Consider *j*th and *k*th trees of *p* and *q*, respectively.
Fix *j*, and note that the observation implies that there exist some natural numbers *q*, then *f* having at least two trees and thus having at least two words that are powers of *b*, which is impossible.
Therefore, σ realizes a bijection from the leaves of *c*.
Since σ is cyclic, the number *c* does not depend on *j*.
We obtain *j*, which, by Lemma 2, implies that the trees

Assume that the

If *i*th tree of *g* can be reduced as a fraction

This implies

Therefore, Thompson’s group *T* has the Haagerup property.

## Remark 1.

Farley constructed a proper cocycle *n* leaves, then

Note that if *V*.
Consider

where

Then

We proved that *T* has the Haagerup property by using the net of maps *V*.
Unfortunately, the maps *V*.
Indeed, consider the sequence of trees *n* leaves.
Let *g* tends to infinity in *V*.

**Funding source: **H2020 European Research Council

**Award Identifier / Grant number: **669240 QUEST

**Funding source: **Australian Research Council

**Award Identifier / Grant number: **DP140100732

**Funding statement: **A. Brothier was supported by European Research Council Advanced Grant 669240 QUEST and is now supported by a UNSW Sydney starting grant.
V. F. R. Jones is supported by the grant numbered DP140100732, Symmetries of subfactors.

## Acknowledgements

We thank the generous support of the New Zealand Mathematics Research Institute and the warm hospitality we received in Raglan, which made this work possible.

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**Received:**2018-05-08

**Revised:**2019-03-14

**Published Online:**2019-05-07

**Published in Print:**2019-09-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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