Haagerup property for wreath products constructed with Thompson's groups

Using recent techniques introduced by Jones we prove that a large family of discrete groups and groupoids have the Haagerup property. In particular, we show that if G is a discrete group with the Haagerup property, then the wreath product $\oplus_{Q_2}G\rtimes V$ obtained from the group G and the usual action of Thompson's group V on the dyadic rational $Q_2$ of the unit interval has the Haagerup property.


Introduction
In the 1930s Ore gave necessary and sufficient conditions for a semi-group to embed in a group, see [Mal53]. Similar properties can be defined for categories giving a calculus of fractions and providing the construction of a groupoid (of fractions) and in particular groups, see [GZ67]. Richard Thompson's groups F Ă T Ă V arise in that way by considering certain diagrammatic categories of forests, see [Bro87,CFP96] and [Bel04,Jon18] for the categorical framework. Recently, Jones discovered a very general process that constructs a group action (called Jones' action) π Φ : G C ñ X Φ from a functor Φ : C Ñ D where C is a category admitting a calculus of fractions and where G C is the group of fractions associated to C (and a fixed object) [Jon17,Jon18], see also the survey [Bro20]. The action remembers some of the structure of the category D and, in particular, if the target category is the category of Hilbert spaces (with linear isometries for morphisms), then π Φ is a unitary representation (in that case we call it a Jones' representation). This provides large families of unitary representations of the Thompson's groups [BJ19b,BJ19a,ABC21,Jon21,BW22]. Certain coefficients of Jones' representations can be explicitly computed via algorithms which makes them very useful for understanding analytical properties of groups of fractions. This article uses for the first time Jones' machinery for proving that new classes of groups (and groupoids) satisfy the Haagerup property. Haagerup property. Recall that a discrete group has the Haagerup property if it admits a net of positive definite functions vanishing at infinity and converging pointwise to one [AW81], see also the book [CCJJA01] and the recent survey [Val18]. It is a fundamental property having applications in various fields such as group theory, ergodic theory, operator algebras, and K-theory for instance. The Haagerup property is equivalent to Gromov's a-(T)-meanability (i.e. the group admits a proper affine isometric action on a Hilbert space) and, as suggested by Gromov's terminology, it is a strong negation of Kazhdan's Property (T): a discrete group having both properties is necessarily finite [Gr93].
AB is supported by the Australian Research Council Grant DP200100067 and a University of New South Wales Sydney Starting Grant.
One additional motivation to study the Haagerup property is given by a deep theorem of Higson and Kasparov: a group having the Haagerup property satisfies the Baum-Connes conjecture (with coefficients) and in particular satisfies the Novikov conjecture [HK01]. Wreath products. The class of groups with the Haagerup property contains amenable groups and many other since it is closed under taking free products and even graph products [AD13]. However, it is not closed under taking extensions and in particular under taking wreath products. We call wreath product (instead of permutational restricted wreath product) a group of the form Γ≀ X Λ :" ' X Γ¸Λ where Γ, Λ are groups, X is a Λ-set, ' X Γ is the group of finitely supported maps from X to Γ, and the action Λ ñ ' X Γ consists in shifting indices using the Λ-set structure of X. It is notoriously a difficult problem to prove that a wreath product has the Haagerup property or not. Cornulier, Stalder and Valette showed that, if Γ and Λ are discrete groups with the Haagerup property, then so does the wreath product ' gPΛ Γ¸Λ and so does ' gPΛ{∆ Γ¸Λ where ∆ is a normal subgroup of Λ satisfying that the quotient group Λ{∆ has the Haagerup property [CSV12]. See also [Cor18] where the later result was extended to commensurated subgroups ∆ ă Λ. However, no general criteria exists for wreath products like ' X Γ¸Λ where X is any Λset. Moreover, there exist many examples of wreath products ' X Γ¸Λ having relative Kazhdan's property (T) thus not having the Haagerup property even when Γ, Λ have it, see [CSV12]. Thompson groups. There have been increasing results on analytical properties of Thompson's groups F Ă T Ă V : Reznikoff showed that Thompson's group T does not have Kazhdan's Property (T) and Farley proved that V has the Haagerup property [Rez01,Far03]. Independently, the works of Ghys-Sergiescu and Navas on diffeomorphisms of the circle implies that F and T do not have Kazhdan's Property (T) [GhS87,Nav02]. Using Jones' technology, Jones and the author constructed explicit positive definite maps on V . This permitted to give two independent short arguments proving that V does not hat Kazhdan's Property (T) and that T has the Haagerup property [BJ19b]. Wreath products using Thompson's groups. In this article we consider wreath products built from actions of Thompson's groups. More precisely, let Q 2 be the set of dyadic rationals in r0, 1q and consider the usual action V ñ Q 2 . Given any group Γ we may form the wreath product Γ ≀ Q 2 V :" ' Q 2 Γ¸V.
More generally, if θ is an automorphism of Γ we may form the twisted wreath product Γ ≀ θ Q 2 V where the action V ñ ' Q 2 Γ is given by the formula: pv¨aqpxq " θ log 2 pv 1 pv´1xqq papv´1xqq for all v P V, a P ' Q 2 Γ, x P Q 2 .
Using Jones' technology we define in this article a net of coefficients vanishing at infinity on the larger group V and thus reproving Farley's result. By mixing these coefficients together with representations of a given group Γ (see below for details) we manage to prove the following result.
Theorem A. Consider a discrete group Γ and an automorphism of it θ P AutpΓq. If Γ has the Haagerup property, then so does the twisted wreath product Γ ≀ θ New examples. Wreath products obtained in Theorem A were not previously known to have the Haagerup property. Moreover, we provide the first analytic but not geometric proof showing that a wreath product has the Haagerup property. Indeed, previous techniques were based on showing that the group admits a proper isometric action (for example using an action on a space with walls). We thank Adam Skalski for pointing this out.
Note that if Γ is finitely presented, then so does the wreath product by a result of Cornulier [Cor06]. Further, if Γ satisfies the homological (resp. topological) finiteness property of being of type F P m (resp. F m ) for any m ě 1 or m " 8, then so does the wreath product Γ ≀ Q 2 V by Bartholdi, Cornulier, and Kochloukova [BdCK15], see also [Bro22b,Section 4.3]. We obtain the first examples of finitely presented wreath products (or of any type F m or F P m with m ě 2) that have the Haagerup property for a nontrivial reason that is: the group acting (here V ) is nonamenable and the base space (here Q 2 ) is not finite. We are grateful to Yves de Cornulier for making this observation. Pairwise non-isomorphic examples. Since the class of groups satisfying the Haagerup property is closed under taking subgroups we obtain the same statement in Theorem A when we replace V by the smaller Thompson's groups F and T . Moreover, note that we obtain infinitely many pairwise non-isomorphic new examples. Indeed, we previously proved that if Γ ≀ θ Q 2 V is isomorphic toΓ ≀θ Q 2 V, then there exists an isomorphism β : Γ ÑΓ and h PΓ satisfyingθ " adphq˝βθβ´1, see [Bro22a,Theorem 4.12]. The same conclusion holds when V is replaced by F or T . We were able to prove Theorem A because Γ≀ θ Q 2 V is the fraction group of a certain category to which we can apply efficiently Jones' technology. These specific groups previously appeared independently in two other frameworks. Indeed, Tanushevski considered those as well as Witzel and Zaremsky [Tan16,WZ18]. Note that the approach of Witzel and Zaremsky, known as cloning systems, is a systematisation of a construction due to Brin of the so-called braided Thompson group [Bri07]. We refer the reader to the appendix of [Bro21] for an extensive discussion on these three independent constructions. A similar diagrammatic construction provides the following groups CpC, Γq¸V where C :" t0, 1u N is the Cantor space and CpC, Γq the group of all continuous maps from C to Γ (i.e. the locally constant maps) equipped with the pointwise multiplication. The action V ñ CpC, Γq is the one induced by the classical action V ñ C on the Cantor space. Even if these groups arise similarly from categories than the wreath products of Theorem A we have been unable to understand their analytic properties leading to the following problem.
Problem B. Assume that Γ is a discrete group with the Haagerup property. Is is true that CpC, Γq¸V has the Haagerup property?
We refer the reader to [Bro21] where we extensively study this specific class of groups. Proof of the main result. The proof is made in three steps.
Step one: we construct a family of functors starting from the category of binary symmetric forests (the category for which Thompson's group V is the group of fractions) to the category of Hilbert spaces giving us a net of positive definite coefficients on V . We prove that this net is an approximation of the identity satisfying the hypothesis of the Haagerup property and thus reproving Farley's result that V has the Haagerup property [Far03].
Step two: given any group Γ we construct a category with a calculus of left-fractions whose group of fractions is isomorphic to the wreath product ' Q 2 Γ¸V. Elements of V are described by (equivalence classes) of triples pt, π, t 1 q where t, t are trees with same number of leaves and π a bijection between the leaves of t and leaves of t 1 . For the larger group ' Q 2 Γ¸V we have a similar description with an extra data being a labeling of the leaves of t, t 1 by elements of the group Γ.
Step three: given a unitary representation of Γ and a functor of step one we construct a functor starting from the larger category constructed in step two and ending in Hilbert spaces. This provides a net of coefficients for the wreath product indexed by representations of Γ and functors of step one. We then extract from those coefficients a net satisfying the assumptions of the Haagerup property.
Step two is not technically difficult but resides on the following key observation: given any functor Ξ : F Ñ Gr from the category of forests to the category of groups we obtain, using Jones' machinery, an action α Ξ : F ñ G Ξ of Thompson's group F on a certain limit group G Ξ . In certain cases (for example when Ξ is monoidal) we can extend α Ξ into a V -action. We observe that there exists a category C Ξ whose group of fractions is isomorphic to the semi-direct product G Ξ¸α Ξ V and this observation works more generally whatever the initial category is, see Remark 2.8. Moreover, the category C Ξ and its group of fractions have very explicit forest-like descriptions allowing us to extend techniques built to study Thompson's group V to the larger group of fractions of C Ξ . By choosing wisely the functor Ξ we obtain that the group of fractions of C Ξ is isomorphic to ' Q 2 Γ¸V. This procedure shows that certain semi-direct products G¸V (or more generally G¸G D where G D is a group of fractions) have a similar structure than V (resp. G D ) and thus we might hope that certain properties of V (resp. G D ) that are not necessarily closed under taking extension might still be satisfied by G¸V (resp. G¸G D ). Note that the groups appearing in Problem B arise in that way. The main technical difficulty of the proof of Theorem A resides in steps one and three; in particular in showing that the coefficients are vanishing at infinity. In step one, we define functors Φ : F Ñ Hilb from binary forests to Hilbert spaces such that the image Φptq of a tree t with n`1 leaves is a sum of 2 n operators. We let this operator acting on a vector obtaining a sum of 2 n vectors. To this functor we associate a coefficient for Thompson's group V where a group element described by a fraction of symmetric trees with n`1 leaves is sent to 2 nˆ2n inner products of vectors. We show that if the fraction is irreducible, then most of those inner products are equal to zero implying that the coefficient vanishes at infinity. In step three we adapt this strategy to a larger category where leaves of trees are decorated with element of the group Γ that requires the introduction of more sophisticated functors. This extension of step one is not straightforward. One of the main difficulty comes from the fact that fractions of decorated trees are harder to reduce. For example, there exists a sequence of tree t n with n leaves such that g n t n t n is a reduced fraction where g n has only one nontrivial entry equal to a fix x P Γ (see Section 2.3.1 for notations). If we forget g n , then the fraction t n t n corresponds to the trivial element of Thompson's group F . Therefore, a naive construction of a functor that would treat independently data of trees and elements of Γ cannot produce coefficients that vanishes at infinity since it will send g n t n t n to a nonzero quantity depending only on x.
The argument works identically for countable and uncountable discrete groups Γ. Interestingly, the coefficients of Thompson's group V appearing in step one are not the one constructed by Farley nor the one previously constructed by the author and Jones but coincide when we restrict those coefficients to the smaller Thompson's group T , see Remark 4.8 and the original articles [Far03,BJ19b]. We could have given a single proof showing that if Γ has the Haagerup property, then so is the associated (possibly twisted) wreath product Γ ≀ Q 2 V. Although, for pedagogical reasons we choose to provide several proofs for various groups. This permits to understand easily the scheme of the proof and to appreciate the gap of difficulties between various cases. We thus prove the Haagerup property for F , then for T , then for V , then for Γ ≀ Q 2 V , and finally for a twisted version of it. The largest gaps of technicality resides between T and V and between V and the wreath product. The proof of Theorem A is based on a categorical and functorial approach that is more natural to use for studying groupoids. We present such a groupoid approach allowing now k-ary forests rather than only binary trees. This leads to the following theorem: Theorem C. Consider a triple pΓ, θ, kq where Γ is a group, θ : Γ Ñ Γ an injective morphism, and k ě 2. There exists a unique monoidal category C (see Section 2.3.1) whose objects are the natural numbers and morphisms from n to m are k-ary forests with n roots, m leaves together with a permutation of the leaves and a labelling of the leaves with elements of Γ. Moreover, the composition of morphisms satisfies the relation Y k˝g " pθpgq, e,¨¨¨, eq˝Y k where g P Γ and Y k is the unique k-ary tree with k leaves. If G C is the universal groupoid of C and Γ is a discrete group that has the Haagerup property, then G C has the Haagerup property.
Note that the groups appearing in Problem B corresponds to the category built from the relation Y˝g " pg, gq˝Y for g P Γ.
If G SF k is the universal groupoid of the category of k-ary symmetric forests, then the automorphism group (i.e. the isotropy group) G SF k pr, rq of the object r is isomorphic to the Higman-Thompson group V k,r , see [Hig74,Bro87]. Further, by adding decoration of the leaves with a group Γ and setting θ " id Γ the identity, we obtain that the isotropy group at the object r is isomorphic to the wreath product where V k,r ñ Q k p0, rq is the usual action of Higman-Thompson's group V k,r on the set of k-adic rationals inside r0, rq. If θ is a nontrivial automorphism, then we obtain a twisted wreath product similarly than in the binary case.
Corollary D. Let Γ be a discrete group with the Haagerup property and θ P AutpΓq an automorphism. Denote by Γ ≀ θ Q k p0,rq V k,r the twisted wreath product associated to the usual action V k,r ñ Q k p0, rq and θ for k ě 2, r ě 1. We have that Γ≀ θ Q k p0,rq V k,r has the Haagerup property.
This corollary generalises Theorem A which corresponds to the case k " 2 and r " 1. Apart from the introduction this article contains five other sections and a short appendix. In Section 2 we introduce all necessary background concerning Thompson's groups, groups of fractions and Jones' actions. We then explain how to build larger categories from functors and how their group of fractions are isomorphic to certain wreath products. In Section 3 we provide short and simple proofs that F and T have the Haagerup property by constructing an explicit net of linear isometries and by considering associated positive definite maps. We then easily observe that they vanish at infinity and converge pointwise to 1. In Section 4, we prove that Thompson's group V has the Haagerup property by refining substantially the proofs for F and T but by keeping the same strategy. It is still easy to see that the positive definite maps converge pointwise to 1. Although, it is much harder to show that they vanish at infinity. In Section 5, we prove Theorem A. We explain how to build matrix coefficient on larger fraction groups. We then follow a similar but more technical strategy. In Section 6, we adopt a groupoid approach. We introduce all necessary definitions and constructions that are easy adaptations of the group case. We then prove Theorem C and deduce Corollary D. In a short appendix we provide a different description of Jones' actions using a more categorical language.
Acknowledgement. We warmly thank Sergei Ivanov, Richard Garner and Steve Lack for enlightening discussions concerning category theory. We thank Adam Skalski for making key comments to us regarding the results and techniques used in this article. We are grateful to Yves de Cornulier and Vaughan Jones for very constructive comments on an earlier version of this manuscript and to Dietmar Bisch, Matt Brin and Yash Lodha for their enthusiasm and encouragements. Finally, we thank Christian de Nicola Larsen for pointing out some typos and technical subtelties in an earlier version of the manuscript.

Preliminaries
2.1. Groups of fractions. We say that a category C is small if its collections of objects and morphisms are both sets. The collection of morphisms of C from a to b is denoted by Cpa, bq. If f P Cpa, bq, then we say that a is the source and b the target of f . As usual we compose from right to left, thus the source of g˝f is the source of f and its target the target of g. When we write g˝f we implicitly assume that g is composable with f meaning that the target of f is equal to the source of g. We sometime write gf for g˝f.
2.1.1. General case. We explain how to construct a group from a small category together with the choice of one of its object. We refer to [Jon18] for details on this specific construction and to [GZ67] for the general theory of calculus of fractions. Let C be a small category and e an object of C satisfying: (1) (Left-Ore's condition at e) If p, q have same source e, then there exists h, k such that hp " kq.
(2) (Weak left-cancellative at e) If pf " qf where f has source e, then there exists g such that gp " gq.
We say that such a category admits a calculus of left-fractions in e.
Proposition 2.1. Let G C be the set of pairs pt, sq of morphisms with source e and common target that we quotient by the equivalence relation generated by pt, sq " pf t, f sq. Denote by t s the equivalence class of pt, sq that we call a fraction. The set of fractions admits a multiplication¨such that t s¨t This confers a group structure to G C such that s t is the inverse of t s and thus t t is the identity for all t. We call G C the group of fractions of pC, eq or of C if the context is clear.
Proof. Given two pairs pt, sq, pt 1 , s 1 q as above there exists by Ore's condition at e some morphisms f, f 1 satisfying f s " f 1 t 1 . We write pt, sq f,f 1 pt 1 , s 1 q for the product giving pf t, f 1 s 1 q.
We claim that f t f 1 t 1 only depends on the classes t s and t 1 s 1 . Consider another pair of morphisms g, g 1 satisfying gs " g 1 t 1 and observe that pt, sq g,g 1 pt 1 , s 1 q " pgt, g 1 s 1 q. By Ore's condition at e there exists h, k such that hf s " kgs. Observe that hf 1 t 1 " hf s " kgs " kg 1 t 1 .
By the weak cancellation property at e there exists b such that bhf 1 " bkg 1 . Moreover, since hf s " kgs we have bhf s " bkgs and thus by the weak cancellation property at e there exists a such that abhf " abkg. We obtain the equalities: (1) bhf 1 " bkg 1 ; (2) abhf " abkg. Observe that f t f 1 s 1 " bhf t bhf 1 s 1 " bhf t bkg 1 s 1 by p1q " abhf t abkg 1 s 1 " abkgt abkg 1 s 1 by p2q " gt g 1 s 1 . This proves the claim. The rest of the proposition follows easily.
When C satisfies the property of above for any of its object we say that it admits a calculus of left-fractions. This is then the right assumptions for considering a groupoid of fractions, see Section 6.1. We will be mostly working with categories of forests defined below and refer to [CFP96,Bel04] for more details about this case. Note that those categories satisfy stronger axioms as they are cancellative (right and left) and satisfies Ore's property at any object.
Remark 2.2. We have followed the original conventions appearing in the first articles on Jones' technology. Unfortunately they are different from the more recent articles when we consider right-fractions instead of left-fractions. Note that t s corresponds formally to t´1˝s and is sometime denoted rt, ss. In more recent articles we often write FracpCq for the fraction groupoid of a category C and FracpC, eq rather than G C for the fraction group of C at the object e.
The formal notation permits to check easily the identities t s¨s u " t u by computing pt´1s q˝ps´1˝uq and check that f˝t f˝s " t s by computing pf˝tq´1˝pf˝sq.
2.1.2. Categories of forests and Thompson's groups. Trees and forests. Let F be the category of finite ordered rooted binary forests whose objects are the nonzero natural numbers N˚:" t1, 2,¨¨¨u and morphisms F pn, mq the set of forests with n roots and m leaves. We represent them as diagram in the plane R 2 whose roots and leaves are distinct points in Rˆt0u and Rˆt1u respectively and are counted from left to right starting from 1. For example f " is a morphism from 3 to 6. A vertex v of a tree has either zero or two descendants v l , v r that are placed on the top left and top right, respectively, of the vertex v. The edge joining v and v l (resp. v r ) is called a left-edge (resp. a right-edge). We compose forests by stacking them vertically so that f˝q is the forest obtained by stacking f on top of q where the i-th root of f is attached to the i-th leaf of q. We obtain a diagram in the strip Rˆr0, 2s that we rescale in Rˆr0, 1s. For example, if t " , then f˝t " .
A tree is a forest with one root and conversely a forest with n roots is nothing else than a list of n trees. Thompson's group F . The category F admits a calculus of left-fractions. We consider the object 1 and note that morphisms with source 1 are trees. The associated group of fractions G F is isomorphic to Thompson's group F .
Fraction. By definition, any element g P F can be expressed as a fraction t s where t, s are trees with the same number of leaves say n. Moreover, if t 1 " f˝t and s 1 " f˝s where f is any forest having n roots, then g is also expressed by the fraction t 1 s 1 . Elementary forest. For any 1 ď i ď n we consider the forest f i,n (denoted by f i if the context is clear) the forest with n roots and n`1 leaves where the i-th tree of f i,n has two leaves and all other trees are trivial. We say that f i,n is an elementary forest. Here is an example: Note that every forest is a finite composition of elementary forests.
Notation 2.3. We write T for the collection of all finite ordered rooted binary trees and by Y " f 1,1 the unique tree with two leaves and I the unique tree with one leaf that we call the trivial tree. By tree we always mean an element of T.
Symmetric forests and Thompson's group V . Consider now the category of symmetric forests SF with objects N˚and morphisms SF pn, mq " F pn, mqˆS m where S m is the symmetric group of m elements. We call an element of SF pn, mq a symmetric forest and, if n " 1, a symmetric tree. Graphically we interpret a morphism pp, σq P SF pn, mq as the concatenation of two diagrams. On the bottom we have the diagram explained above for the forest p in the strip Rˆr0, 1s. The diagram of σ is the union of m segments rx i , x σpiq`p 0, 1qs, i " 1,¨¨¨, m in Rˆr1, 2s where the x i are m distinct points in Rˆt1u such that x i is on the left of x i`1 . The full diagram of pp, σq is obtained by stacking the diagram of σ on top of the diagram of p such that x i is the i-th leaf of p. If we consider the permutation τ such that τ p1q " 2, τ p2q " 3, τ p3q " 1, then its corresponding diagram is , then the diagram associated to pt, τ q is .
Two kinds of morphisms. We interpret the morphism pp, σq as the composition of the morphisms pI m , σq˝pp, idq where I m is the trivial forest with m roots and m leaves (thus m trivial trees next to each other) and id is the trivial permutation. By identifying σ with pI m , σq and p with pp, idq we obtain that pp, σq " σ˝p. We have already defined compositions of forests in the description of the category F . The composition of permutations is the usual one. It remains to explain the composition of a forest with a permutation. Consider a permutation τ of n elements and a forest p with n roots and m leaves and let l i be the number of leaves of the i-th tree of p. We define the composition as: where τ ppq is the forest obtained from p by permuting its trees such that the i-th tree of τ ppq is the τ piq-th tree of p and Spp, τ q is the permutation corresponding to the diagram obtained from τ where the i-th segment rx i , x τ piq`p 0, 1qs is replaced by l τ piq parallel segments. For example, if we consider the forest f " and the permutation τ " , then f˝τ " Spf, τ q˝τ pf q where τ pf q " and Spf, τ q " . This is a category admitting a calculus of left-fractions whose group of fractions associated to pSF , 1q is isomorphic to Thompson's group V . Note that the relations between forests and permutations can be interpreted as a Brin-Zappa-Szép product of the category of forests F and the groupoid of all symmetric groups. For more details on such products we refer the reader to the articles of Brin and of Witzel and Zaremsky [Bri07,WZ18].
Elements of V as fractions. Any element of V is an equivalence class of a pair of symmetric trees τ˝t σ˝s . Observe that τ˝t σ˝s " σ´1˝τ˝t s . Hence, any element of V can be written as σ˝t s for some trees t, s and permutation σ. Note that formally the fraction τ˝t σ˝s is equal to the signed path of morphisms pτ˝tq´1˝pσ˝sq " t´1˝τ´1˝σ˝s.
Affine forests and Thompson's group T . Let Z{mZ be the cyclic group of order m identified as a subgroup of the symmetric group S m and consider the subcategory AF Ă SF of affine forests where It is a category admitting a calculus of left-fractions and the group of fractions associated to the objet 1 is isomorphic to Thompson's group T . We will often identify F and AF as subcategories of SF giving embeddings at the group level F Ă T Ă V . Reduced pair. We say that a pair of symmetric trees pτ˝t, σ˝sq is reduced if there are no other pairs pτ 1˝t1 , σ 1˝s1 q in the same class such that t 1 has strictly less leaves than t. Monoidal structure. We equipped SF with a monoidal structure b that is for objects n, m and the tensor product of two symmetric forests consists in concatenating the two diagrams horizontally such that pσ˝f q is placed to the left of pσ 1˝f 1 q. If we consider the tree and forest t, f of above, then This monoidal structure of SF confers a monoidal structure on the smaller category F but not on AF as a product of cyclic permutations is in general not a cyclic permutation.
Remark 2.4. Note that the common definition of a monoidal or tensor category demands that b has a neutral element. Here, this can be added by considering the object 0 and the empty diagram playing the role of id 0 .
Metric. We equip forests with the usual metric. Hence, an edge between two vertices if of length one. Now, recall that by convention the trivial tree I has one root and one leaf that are equal and thus is of diameter zero. If Y is the tree with two leaves, then each of its leaf is at distance one from the root. If we consider the tree t " , then its first leaf is a distance two from the root and the second and third leaves are at distance two and one from the root, respectively. Order. We equip F with a partial order ď defined as follows: Note that if s, t are trees, then s ď t if and only if s is a rooted subtree of t. Moreover, the set of trees equipped with ď is directed, i.e. for all trees s, t there exists a third tree z satisfying that s ď z and t ď z.

Classical actions of the Thompson's groups on the unit interval.
We present the usual action of V on the unit interval which explains the correspondence between trees and certain partitions of the unit interval. Additional details can be found in [CFP96]. Standard dyadic interval and partition. Consider the infinite binary rooted tree t 8 and decorate its vertices by intervals such that the root corresponds to the half-open interval r0, 1q and the successors of a vertex decorated by rd, d 1 q are decorated by rd, d`d 1 2 q to the left and r d`d 1 2 , d 1 q to the right. Here is the beginning of this labelled tree: Intervals appearing in this tree are called standard dyadic intervals and form the set tr a 2 n , a`1 2 n q : n ě 0, 0 ď a ď 2 n´1 u. Consider a tree t P T and write I n for the interval corresponding to the n-th leaf of t where t is viewed as a rooted subtree of t 8 . We have that tI 1 ,¨¨¨, I n u is a partition of r0, 1q that we call a standard dyadic partition.
Action of V on the unit torus. Now consider g " τ˝t σ˝s P V and the standard dyadic partitions tI 1 ,¨¨¨, I n u and tJ 1 ,¨¨¨, J n u of r0, 1q associated to the trees s and t respectively. The element g acting on r0, 1q is the unique piecewise linear function with positive constant slope on each I k that maps I σ´1piq onto J τ´1piq for any 1 ď i ď n. From this description of V ñ r0, 1q we easily deduce that T is the group of homeomorphisms of the unit torus that is piecewise affine with slopes powers of 2 and finitely many breakpoints while F is the subgroup of T fixing 0 (and thus acting on r0, 1s by homeomorphisms). Action of V on the dyadic rationals. Put Q 2 the set of dyadic rational in r0, 1q and observe that the action of V on r0, 1q restricts to an action on Q 2 . This action will appear in the construction of the wreath product ' Q 2 Γ¸V of the main theorem. Note that the action V ñ Q 2 is conjugated to the homogeneous action of V ñ V {V 1{2 where V 1{2 is the stabiliser subgroup of the point 1{2.

Jones' actions.
2.2.1. General case. Consider a small category C admitting a calculus of left-fractions in a fixed object e, another category D whose objects are sets, and a covariant functor Φ : C Ñ D. Consider the set of morphisms with source e that we equip with the following order: t ď s if there exists f satisfying s " f˝t. This is the generalisation of the order we put on the set of trees at the end of Section 2.1.2. Note that it is a directed set precisely because C satisfies Ore's condition in e. Given t P Cpe, bq, we form the set X t a copy of Φpbq and consider the directed system pX t : t a morphism with source eq with maps ι f t t : X t Ñ X f t given by Φpf q. Let X be the inductive limit that we write lim Ý Ñt,Φ X t to emphasize the role of Φ. It can be described as tpt, xq : t P Cpe, bq, x P Φpbq, b P obpCqu{ " where " is the equivalence relation generated by pt, xq " pf t, Φpf qpxqq.
We often denote by t x the equivalence class of pt, xq and call it a fraction.
Definition 2.5. Let G C be the group of fractions of C at the object e.The Jones action π Φ : G C ñ X associated to the functor Φ : C Ñ D is defined by the following formula: π Φˆt s˙r x :" pt Φpqqpxq for p, q satisfying ps " qr.
One can check that this formula does not depend on the choice of p, q and thus the action is well-defined.
(1) When C is right-cancellative at e and t ď s, then there exists a unique f satisfying s " f t. Although, when C is only weak right-cancellative at e, then there may be several f satisfying s " f t. We still obtain a directed system but to stay fully rigorous we should write ι t,f rather than ι s t since there may be several maps going from X t to X s .
(2) Note that if C admits a calculus of left-fractions (at any objects), then we can adapt the construction and obtaining an action of the whole groupoid of fractions, see Section 6.
(3) If we replace X t by the set of morphisms DpΦpeq, Φptargetptqqq in the construction, then we no longer need to assume that the objects of the category D are sets. This was the original definition of Jones [Jon18]. (4) A similar construction can be done for contravariant functors Φ : C Ñ D leading to an action of G C . Formally, this makes no difference since we may consider the opposite category of D and recovering a covariant functor. Although, in practice we will obtain inverse systems and limits rather than direct systems and colimits. For instance, if D is the category of finite groups, then a covariant functor will typically provide an amenable discrete group while a contravariant functor will provide a profinite group.

2.2.2.
The Hilbert space case: representations and coefficients. Let D " Hilb be the category of complex Hilbert spaces with linear isometries for morphisms. Consider a functor Φ : C Ñ Hilb. We often write H t " X t for the Hilbert space associated to t P Cpe, bq. The inductive limit has an obvious pre-Hilbert space structure that we complete into a Hilbert space and denote by H Φ " lim Ý Ñt,Φ H t . The Jones action π Φ : G C ñ H Φ is a unitary representation that we call a Jones' representation.
Let H be the Hilbert space Φpeq associated to the chosen object e that we consider as the subspace H id of H Φ where id P Cpe, eq is the identity morphism. Note that if ξ is a vector of H and g " t s P G C is a fraction, then (2.1) xπ Φˆt s˙ξ , ξy " xΦpsqξ, Φptqξy.
We will be considering exclusively those kind of coefficients that can be easily computed if one understand well the functor Φ. In particular, if Φpnq is a space constructed via a planar algebra, like in [Jon17, ABC21, Jon21], then the coefficient of above can be computed using the skein theory of the planar algebra giving us an explicit algorithm, see also [Ren18,GS15].
2.2.3. The group case. Let D " Gr be the category of groups and consider a functor Φ : C Ñ Gr . We often write Γ t " X t for the group associated to a morphism t P Cpe, bq.
The inductive limit lim Ý Ñt,Φ Γ t is usually denoted G Φ and has a group structure. Moreover, the Jones' action π Φ : G C ñ G Φ is an action by group automorphisms. We equipped Gr with the monoidal structure b such that Γ 1 b Γ 2 is the direct sum of these groups. If σ i : Γ i Ñ Λ i , i " 1, 2 are group morphisms, then σ 1 b σ 2 is the following group morphism Functors of this form were first considered by Stottmeister and the author in [BS19a,BS19b]. A systematic study of the semi-direct product of groups G Φ¸GC has been initiated in [Bro22a,Bro21].
2.2.4. Monoidal functors. We will mainly consider covariant monoidal functors from the category of forests F into Hilb or Gr . On Hilb we consider in this article the classical monoidal structure b so that ℓ 2 pIq b ℓ 2 pJq » ℓ 2 pIˆJq. Observe that an elementary forest f i,n decomposes as follows Φpnq " Φp1q bn and Φpf i,n q " id bi´1 bΦpY q b id n´i . Since any forest is the composition of some f i,n we obtain that Φ is completely characterized by the objet Φp1q and the morphism ΦpY q : Φp1q Ñ Φp1q b Φp1q. When D " Hilb we may use the following notations: H :" Φp1q and R :" ΦpY q. In that case R : H Ñ H b H is a linear isometry. If D " Gr, then we may adopt the notations: Ξ : F Ñ Gr with Γ :" Ξp1q and S :" ΦpY q. Hence, S : Γ Ñ Γ ' Γ is a group morphism. Given a monoidal functor Φ : F Ñ D we have a Jones' action π Φ : F ñ X . Assume that D is a symmetric category like Hilb and Gr. We can then extend this action into an action of the larger Thompson's group V via the formula When D " Hilb, then the formula (2.1) becomes: xπ Φˆθ˝t σ˝s˙ξ , ξy " xTenspσqΦpsqξ, TenspθqΦptqξy for ξ P Φp1q.
Here is another interpretation of the extension of the Jones action to Thompson's group V . We extend the monoidal functor Φ : F Ñ D uniquely into a monoidal functor Φ : SF Ñ D satisfying Φp1q " Φp1q, ΦpY q " ΦpY q and where Φpσq " Tenspσq for a permutation σ.
We then perform the Jones construction applied to Φ. We have an inductive limit of spaces H σ˝t where now Hilbert spaces are indexed by pairs pσ, tq with t a tree and σ a permutation. Observe that H σ˝t embeds inside H t via Φpσ´1q and thus the limit Hilbert space for the functor Φ can be canonically identified with the one of Φ since any morphism of SF with source 1 (a symmetric tree) is smaller than a morphism of F with source 1 (a tree), i.e. the set of trees is cofinal inside the directed set of symmetric trees. The Jones action for Φ of the larger group of fractions G SF satisfies that as in (2.2).
2.3. Construction of larger groups of fractions. This section explains how to achieve step 2 described in the introduction: given a functor Ξ : F Ñ Gr we construct a category C Ξ whose group of fractions is isomorphic to the semi-direct product G¸V where V ñ G is the Jones action induced by Ξ.
2.3.1. Larger groups of fractions. A functor gives an action. Consider a group Γ, a group morphism S : Γ Ñ Γ ' Γ, and the unique monoidal functor Ξ : F Ñ Gr satisfying that Ξp1q " Γ and ΞpY q " S. Set G :" lim tPT,Ξ Γ t the inductive limit group with respect to (w.r.t.) this functor where Γ t :" tpg, tq, g P Ξptargetptqqu is isomorphic to Γ n when t is a tree with n leaves. Intuitively, Γ t can be interpreted as all possible decorations of the leaves of t with elements of Γ. We have a Jones' action π Ξ : F ñ G that we extend to an action π Ξ : V ñ G as explained above. Since π Ξ is an action by group automorphisms we can construct the semi-direct product G¸π Ξ V. Group of fractions. We now show that G¸π Ξ V arises naturally as a group of fractions. Define the category C :" C Ξ with object N˚and sets of morphisms Cpn, mq :" F pn, mqˆS mˆΓ m .
We interpret F pn, mq (resp. S m and Γ m ) as morphisms in Cpn, mq (resp. in Cpm, mq), i.e. a triple pf, σ, gq P Cpn, mq is interpreted as a composition g˝σ˝f. A morphism is identified with an isotopy class of diagrams that are vertical concatenation of forests, permutations, and a tuple of elements of Γ. Composition of morphisms. We previously explained what are the diagrams for forests and permutations and how to compose permutations with forests. We now explain how to compose tuples of elements of Γ with forests and permutations. An element g " pg 1 ,¨¨¨, g m q P Γ m is the diagram consisting of placing n dots on a horizontal line labeled from left to right by g 1 , g 2 ,¨¨¨, g m . If f P F pn, mq, then the diagram g˝f is represented by the forest f whose j-th leaf is labeled by g j . If f " and g " pg 1 , g 2 , g 3 q, then g˝f " If p P F pm, kq is another forest, then the diagram p˝g is represented by the forest p whose j-th root is labeled by g j . For example, if p " , then p˝g " . Now, we can lift up the g i 's on top of the forest p by applying the functor Ξ. We obtain that p˝g " Ξppqpgq˝p. The element Ξppqpgq is an element of Γ 6 which decorates the six leaves of the forest p. This process shows that a forest (here p) with roots decorated by elements of Γ is equal to the same forest with now its leaves decorated by elements of Γ. Formally, the rules of compositions are: f˝g :" Ξpf qpgq˝f, @f P F pn, mq, g P Γ n σ˝pg 1 ,¨¨¨, g n q " pg σ´1p1q ,¨¨¨, g σ´1pnq q˝σ, @g i P Γ, σ P S n This indeed defines associative compositions for morphisms and provides a categorical structure to C. Define a monoidal structure b on C such as n b m :" n`m for objects and the tensor product of morphisms corresponds to horizontal concatenation from left to right as in SF . The following proposition follows from the definitions of calculus of left-fractions.
Proposition 2.7. The category C admits a calculus of left-fractions. Its group of fractions G C associated to the object 1 is isomorphic to the semi-direct product G¸π Ξ V constructed via the functor Ξ : F Ñ Gr .
Proof. The two axioms of calculus of left-fractions are trivially satisfied by C. Let us build an isomorphism from G¸π Ξ V to G C . Consider v P V and g P G . There exists a large enough tree t such that v " t σs and g P Γ t where s is another tree and σ a permutation. To emphasise that we consider the representative of g inside Γ t we write g as a fraction t g t . Define the family of maps: Those maps are compatible with the directed systems associated to V, G , and Our maps satisfy the following: The limit map lim Ý Ñt P t defines a group isomorphism from G¸π Ξ V onto G C .
Fractions. Every element of V can be written as a fraction σt s where t, s are trees with the same number of leaves and σ is a permutation. Similarly, using composition of morphisms inside the category C Ξ , we observe that any element of G C can be written as a fraction σgt s " gt σ´1s like in V but where we labeled the leaves of t with elements of the group Γ.
Remark 2.8. We have explained how to construct a category C Ξ from a functor Ξ : F Ñ Gr starting from the category of forests such that the group of fractions of C Ξ is isomorphic to the semi-direct product obtained from the Jones action induced by Ξ. This process is very general and we can replace the category F by any other small category D admitting a calculus of left-fractions at a certain object e P obpDq. Indeed, consider a functor Ξ : D Ñ Gr and the associated Jones' action α Ξ : G D ñ G Ξ where G D is the group of fractions of pD, eq. Define a new category C Ξ with object obpC Ξ q " obpDq and morphisms C Ξ pa, bq " Dpa, bqˆΞpbq for a, b objects. As before we identify Dpa, bq and Ξpbq as morphisms of C Ξ from a to b and from b to b respectively. The composition of morphisms of C Ξ are defined such that f˝g " Ξpf qpgq˝f, for f P Dpa, bq, g P Ξpaq, a, b P obpC Ξ q.
One can check that C Ξ is a small category admitting a calculus of left-fractions at e whose associated group G C Ξ is isomorphic to the semi-direct product G Ξ¸GD . In particular, we can choose to replace permutations by braids and obtaining braided versions of our groups. This produces wreath product where the braided Thompson group is acting rather than V .
Notation 2.9. We often write v for an element of V , g for an element of Γ or Γ n and v g for an element of G C .
Extending Jones' actions to larger categories. We explain how to extend a Jones' action to a larger category. Assume we have a monoidal functor Φ : F Ñ D into a symmetric category. This defines a Jones' action π : F ñ X that can be extended to an action of V as we saw in Section 2.2.4. Let us explain how this same process allow us to extend π to an action of the even larger group G C where C " C Ξ . Write X :" Φp1q and assume we have an action by automorphisms ρ : Γ ñ X. We extend π to the group of fractions G C such as: for t, s trees with n leaves, σ P S n and g P Γ n . Formula 2.3 can be obtained as follows. Extend the functor Φ into a functor Φ : C Ñ D such that Φp1q " Φp1q, ΦpY q " ΦpY q and Φpσq " Tenspσq, Φpgq " ρpgq, σ P S n , g P Γ. We observe that for any morphism gσt of C with source 1 we have that gσt ď t and thus we can identify the inductive limit X obtained with Φ with the inductive limit obtained with Φ. Therefore, which recovers Formula 2.3.

2.3.2.
Isomorphism with a wreath product. We end this subsection by giving a precise description of G C for a specific choice of functor. Let V ñ Q 2 be the restriction of the usual action of V on the unit interval to the dyadic rationals Q 2 , see Section 2.1.3 for details. Let Γ be a group and θ P AutpΓq an automorphism of Γ. Given v P V and x P Q 2 we write v 1 pxq for the right-derivative of v at x. Moreover, we denote by log 2 the logarithm in base 2 so that log 2 p2 n q " n for all n P Z. Consider the direct sum ' Q 2 Γ of all maps a : Q 2 Ñ Γ that are finitely supported and define the actions We write Γ ≀ θ Q 2 V :" ' Q 2 Γ¸θ V for the associated semi-direct product that we call a twisted wreath product. When θ " id is the identity we drop the superscript θ and say that we have a wreath product or an untwisted wreath product. Here is a key observation that was done in [Bro22a, Section 4.2].
Denote by G :" lim Ý ÑtPT,Ξ Γ t the limit group obtained and by π Ξ : V ñ G the Jones action. There is a group isomorphism from G onto ' Q 2 Γ that intertwines the Jones action π Ξ : V ñ G and the twisted action V ñ ' Q 2 Γ described above. In particular, the group of fractions G C associated to the larger category C :" C Ξ is isomorphic to the twisted wreath product Γ ≀ θ Q 2 V . Note that it is easy to understand graphically the composition of morphisms in the category C Ξ associated to the specific functor Ξ of Proposition 2.10. Indeed, Y˝g " pθpgq, eq˝Y for any g P Γ. Hence, elements of Γ can go up in a tree by going to the left and by adding some trivial elements e to their right. For example, if g " pg 1 , g 2 , g 3 q and f " , then f˝g " g 1 g 2 g 3 " Ξpf qpgq˝f " g 1 θpg 2 q e θpg 3 q e e .

Haagerup property for Thompson's groups F and T
In this article we prove that certain wreath products have the Haagerup property. This result is new and is done by using the original definition of the Haagerup property: there exists a net of positive definite maps vanishing at infinity that converges pointwise to 1. The construction of the net is done using Jones' technology and by identifying wreath products with certain groups of fractions. We could give a single proof. However, for pedagogical reasons we will give five of them with increasing level of technicality. More precisely, we provides proofs for the following results: (1) Thompson's group F has the Haagerup property; (2) Thompson's group T has the Haagerup property; (3) Thompson's group V has the Haagerup property; (4) If Γ has the Haagerup property, then so does the wreath product Γ ≀ Q 2 V ; (5) If Γ has the Haagerup property and θ P AutpΓq is any automorphism of Γ, then the associated twisted wreath product Γ ≀ θ Q 2 V has the Haagerup property. The important gaps of difficulties between these cases are from T to V and from V to the untwisted wreath product.
3.1. Proof for Thompson's group F . Consider the Hilbert space H :" ℓ 2 pNq where N is the additive monoid of natural numbers (including zero). We write pδ n : n ě 0q for the usual orthonormal basis of H. We identify H bk with ℓ 2 pN k q and consider the usual orthonormal basis pδ x : x P N k q of it for all k ě 1. Fix a real number 0 ď α ď 1 and set β :" ? 1´α 2 . We now define a linear isometry: This defines uniquely a monoidal covariant functor Φ α : F Ñ Hilb and thus a Jones' representation π α : F ñ H α . Now, H embeds in H α and we may then consider δ 0 as a unit vector of H α . We set φ α : F Ñ C, g Þ Ñ xπ α pgqδ 0 , δ 0 y our matrix coefficient which is a positive definite map. Key fact. Consider a tree t with n leaves and the list d t :" pd t 1 ,¨¨¨, d t n q of distances between the root of t and each of its leaf. The map t Þ Ñ d t is injective. With this fact we will be able to easily prove the Haagerup property for F . By the key fact we have that when α " 0, then the cyclic component of π 0 associated to the vector δ 0 is unitary equivalent to the left-regular representation λ F : F ñ ℓ 2 pF q. When α " 1, then the cyclic component of δ 0 becomes unitary equivalent to the trivial representation 1 F . Hence, we have constructed a continuous path of representations between the trivial and the left-regular ones. In particular, for all g P F we have that lim αÑ1 φ α pgq " 1. To conclude that F has the Haagerup property it is then sufficient to prove that for all 0 ă α ă 1 we have that φ α vanishes at infinity. We explain briefly why this is the case.
Consider g " t s in F where t, s are trees with same number of leaves say n. Observe that φ α pgq " xπ α p t s qδ 0 , δ 0 y " xΦ α psqδ e , Φ α ptqδ e y.
The vector Φ α psqδ e belongs to H bn and can easily be decomposed over the usual orthonormal basis. Indeed, for each rooted subtree x of s we realise the decomposition s " f x˝x where f x is a uniquely defined forest. The forest f x has n leaves. We write d x,s j for the distance from this j-th leaf of f to the root of f x that is in the same connected component.
where d x,s is the multi-index pd x,s 1 ,¨¨¨, d x,s n q and c x a certain coefficient equal to a product of α and β. Similarly, Φ α ptqδ e admits such a decomposition into ř y c y δ d y,t . Therefore, Observe that xδ d x,s , δ d y,t y " 1 when d x,s " d y,t meaning that the forests f x and f y are equal by the key fact of above.
We deduce the following second key fact: if t s is an irreducible fraction we have that all the coefficients of above are equal to zero except one: the coefficient corresponding to the subtrees x " s and y " t implying that f x " f y " I bn are trivial. Indeed, if there would be another nonzero coefficient then there would exists proper subtrees x ď s, y ď t so that f x " f y ‰ I bn . This implies that t s can be reduced into y x and thus contradicting our assumption of irreducibility. We deduce that φ α pgq " α 2n´2 for g equal to an irreducible fraction made of trees with n leaves. Since there are only finitely many of those for each fixed n we deduce that φ α vanishes at infinity for all 0 ď α ă 1 and thus F has the Haagerup property. Note that π α extends canonically into a representation of V . However, φ α is no longer vanishing at infinity when extended to V nor on the intermediated subgroup T . Indeed, if g n " t n˝σ t n where t n is the regular tree with 2 n leaves all at distance n from the root and σ is a n-cycle, then φ α pg n q " 1 for all n and α.

3.2.
Proof for Thompson's group T . We proceed similarly than in the F -case. Instead of considering N we consider the free monoid M " N˚N in two generators a, b. We write e for the trivial element of M. As above we write H " ℓ 2 pMq for the associated Hilbert space and pδ x : x P Mq for the usual orthonormal basis. Fix 0 ď α ď 1, set β :" ? 1´α 2 , and define the linear isometry: This provides a functor Φ α , a Jones representation π α : T ñ H α , and a matrix coefficient: We have that the cyclic subrepresentation of π α associated to the vector δ e interpolates the trivial and the left-regular representations of T . To obtain the Haagerup property for T it is then sufficient to show that φ α vanishes at infinity for all 0 ă α ă 1. Key fact: Consider a tree t with n leaves and σ a cyclic permutation of t1,¨¨¨, nu. We write w t i for the (unique geodesic) path from the root of t to its i-th leaf. We identify w t i with a word x 1¨¨¨xk in the letters a, b where k is the length of the path and x j " a when the j-th edge of the path is a left-edge and x j " b otherwise. The map pt, σq Þ Ñ pw t σp1q ,¨¨¨, w t σpnq q is injective.
Using the key fact we can proceed similarly than above and conclude that if g " t˝σ s P T is a reduced fraction with t, s trees with n leaves, and σ a cyclic permutation, then φ α pgq " α 2n´2 .
This proves that T has the Haagerup property. Note that when we extend φ α to V we no longer have a map vanishing at infinity, see [BJ19b, Remark 1].

Haagerup property for Thompson's group V
4.1. The family of isometries, functors, representations, and matrix coefficients. Consider the free monoid M in the four generators a, b, c, d and let H :" ℓ 2 pMq be the associated Hilbert space with usual orthonormal basis pδ x : x P Mq. Note that we use the free monoids in one, two, and four generators for constructing matrix coefficients for F, T, and V , respectively. Identify H bn with ℓ 2 pM n q and thus the standard orthonormal basis of H bn consists in Dirac masses δ w where w is a list of n words in letters a, b, c, d. For any real number 0 ď α ď 1 we set β :" ? 1´α 2 and define the isometry Let Φ α : F Ñ Hilb be the associated monoidal functor satisfying Φ α p1q :" H, Φ α pY q " R α and let π α : V Ñ UpH α q be the associated Jones' representation. Define the coefficient φ α : V Ñ C, v Þ Ñ xπ α pvqδ e , δ e y.

Interpolation between the trivial and the left-regular representations.
It is easy to see that the representations π 0 and π 1 , that we restrict to the cyclic space generated by δ e , are unitary equivalent to the left-regular representation λ V and to the trivial representation 1 V , respectively. In particular, lim αÑ1 φ α pvq " 1 for any v P V . By definition, φ α is positive definite for any α. Therefore, it is sufficient to show that φ α vanishes at infinity for any 0 ă α ă 1 to prove that V has the Haagerup property. From now on we fix 0 ă α ă 1 and suppress the subscript α thus writing R, Φ, π, φ for R α , Φ α , π α , φ α .

4.3.
The set of states. Consider a tree t P T with n leaves. Put Vptq the set of trivalent vertices of t that is a set of order n´1 and let Stateptq :" tVptq Ñ t0, 1uu be the set of maps from the trivalent vertices of t to t0, 1u that we call the set of states of t. Consider the maps By definition we have R " Rp0q`Rp1q.
Given a state τ P Stateptq, we consider the operator Rpτ q : H Ñ H bn defined as follows.
If t decomposes as a product of elementary forests f j n´1 ,n´1˝fj n´2 ,n´2˝¨¨¨fj 2 ,2˝f1,1 and if ν k is the unique trivalent vertex of f j k ,k , then Rpτ q " pid bj n´1´1 bRpτ pν n´1 qq b id n´1´j n´1 q˝¨¨¨˝Rpτ pν 1 qq.
By definition of the functor Φ we obtain the formula Φptq " ÿ τ PStateptq Rpτ q.
When applied to δ e we obtain: where α τ is a constant depending on the state τ and W pt, τ q is a list of words of M (one word per leaf). For example, if t is the tree of the figure of above, then we have four coefficients corresponding to the states taking the values p0, 0q, p0, 1q, p1, 0q, and p1, 1q at the pair of vertices pν 1 , ν 2 q. We obtain: Φptqδ e " α 2 δ e,e,e`α βδ c,d,e`β αδ ca,cb,d`β 2 δ cc,cd,d .
If σ P S n is a permutation, then where σW pt, τ q is the list of words permuted by σ.

4.4.
General strategy for proving that φ vanishes at infinity. Consider a fraction v " σ˝t t 1 . The decomposition of above provides the following: If t has n`1 leaves, then the coefficient of above is a sum of 2 nˆ2n inner products of vectors. Our strategy is to prove that most of them are equal to zero when σt t 1 is a reduced fraction, i.e. σW pt, τ q ‰ W pt 1 , τ 1 q for most pairs of states pτ, τ 1 q. Let us describe the j-th word W pt, τ q j of W pt, τ q. Consider the j-th leaf ℓ of the tree t and let P j be the geodesic path from the root of t to this leaf. Denote by ν 1 ,¨¨¨, ν k the trivalent vertices of this path listed from bottom to top and let e 1 ,¨¨¨, e k be the edges such that the source of e i is ν i and its target ν i`1 for 1 ď i ď k´1 while e k goes from ν k to the leaf ℓ. We have e if τ pν 1 q "¨¨¨" τ pν i q " 0 a if e i is a left-edge and τ pν i q " 0 c if e i is a left-edge and τ pν i q " 1 b if e i is a right-edge and τ pν i q " 0 d if e i is a right-edge and τ pν i q " 1 when in the second and fourth case we further assume that at least one of the τ pν j q is equal to 1 for 1 ď j ă i. From this description we easily deduce the following lemma.
Observe that if r :" maxpi : τ pν s q " 0 for all s ď iq, then W pt, τ q j " ypr`1qypr2 q¨¨¨ypkq with ypr`1q " c or d. Further, Equation 4.4 shows that the word W pt, τ q j remembers the part of the path after the r`1-th vertex. This motivates the following decomposition.
Notation 4.2. If τ is a state of the tree t, then we define z τ to be the largest rooted subtree of t satisfying that τ pνq " 0 for all (trivalent) vertices ν of z τ (hence excluding the leaves of z τ ). Denote by f τ the unique forest satisfying that t " f τ˝zτ .
Key observation: The list of words W pt, τ q remembers the forest f τ , i.e. if t is a fixed tree and τ, τ 1 are two states on two different trees t, t 1 , then W pt, τ q " W pt 1 , τ 1 q implies that f τ " f τ 1 .

4.5.
An equivalence relation on the set of vertices. From now on we consider an element v P V that we decompose as a fraction v " σ˝t t 1 where t, t 1 are trees with n leaves and σ is a permutation that we interpret as a bijection from the leaves of t to the leaves of t 1 . We define an equivalence relation on the set of trivalent vertices of the tree t which depends on the triple pt, t 1 , σq.
Definition 4.3. Consider two trivalent vertices ν,ν of t. Assume that there exists a trivalent vertex ν 1 of t 1 and two leaves ℓ,l of t that are descendant of ν,ν, respectively, and satisfying: (1) the leaves σpℓq and σplq are descendant of ν 1 ; (2) dpν, ℓq " dpν 1 , σpℓqq and dpν,lq " dpν 1 , σplqq where d is the usual distance on trees. In that case we say that ν is equivalent toν and write ν "ν.
It is easy to see that " defines an equivalence relation. The next proposition implies that there are very few pairs of states pτ, τ 1 q satisfying that W pt 1 , τ 1 q " σW pt, τ q.
Proof. Proof of (1). Consider vertices ν,ν of t that are equivalent under the relation ". Denote by ℓ,l and ν 1 as in Definition 4.3. The equality σW pt, τ q " W pt 1 , τ 1 q together with Formula 4.4 imply that τ pνq " τ 1 pν 1 q and τ pνq " τ 1 pν 1 q. Proof of (2). Assume that ν is a vertex of f τ and that there are no otherν such that ν "ν. We will show that the fraction σ˝t t 1 is necessarily reducible. Let t ν be the maximal subtree of t with root ν. Hence, the leaves of t ν are all the leaves of t that are descendant of ν. Note that since ν is a trivalent vertex we have that the tree t ν has at least two leaves (and is thus nontrivial). For each leaf ℓ of t ν we consider c ℓ : the geodesic path from ν to ℓ. Consider now the leaf σpℓq of t 1 and c 1 ℓ the geodesic path in t 1 ending at σpℓq and of same length than c ℓ . The equality σW pt, τ q " W pt 1 , τ 1 q implies that the distance between ℓ and a root of f τ is equal to the distance between σpℓq and a root of f τ 1 . Since ν is a vertex of f τ , the whole path c ℓ is contained in f τ , and therefore the whole path c 1 ℓ is contained in f τ 1 . Denote by s 1 the subgraph of t 1 equal to the union of all the paths c 1 ℓ where ℓ runs over all the leaves of t ν . We are going to show that s 1 is a tree isomorphic to t 1 . We claim that all the paths c 1 ℓ starts at a common vertex ν 1 of t 1 . Indeed, denote by V 1 the set of all the sources of the paths c 1 ℓ . Let f 1 Ă t 1 be the maximal subforest whose set of roots is equal to V 1 . If ℓ 1 is a leaf of f 1 , then we can consider σ´1pℓ 1 q which is a leaf of t.
By assumption there are no otherν in t that is equivalent to ν. This forces to have that σ´1pℓ 1 q is a leaf of t ν for all leaf ℓ 1 of f 1 . Moreover, by repeating this argument we deduce that all leaves of t ν must be equal to a certain σ´1pℓ 1 q with ℓ 1 a leaf of f 1 , i.e. σ restricts to a bijection from the leaves of t ν to the leaves of f 1 . By using that f 1 Ă f τ 1 and t ν Ă f τ we deduce by an induction on the number of leaves of t ν that f 1 must be a tree that we write t 1 ν . This uses that W pt, τ q remembers the forest f τ and in particular the structure of subforests of it like t ν . This proves the claim. Hence, all c 1 ℓ starts at a common vertex ν 1 of t 1 . The equality σW pt, τ q " W pt 1 , τ 1 q together with the fact that t ν Ă f τ and f 1 Ă f τ 1 implies (via an easy induction on the number of leaves of t ν ) that σ respects the order of the leaves, i.e. the i-th leaves of t ν is sent by σ to the i-th leaf of t 1 ν for any i. Using again the equality σW pt, τ q " W pt 1 , τ 1 q we deduce that the two trees t ν and t 1 ν are necessarily isomorphic (as ordered rooted binary trees). This implies that we can reduce the fraction σ˝t t 1 by removing t ν and t 1 ν at the numerator and denominator. Since t ν was supposed to be nontrivial we obtain that our fraction σ˝t t 1 is reducible, a contradiction. Proof of (3). By Lemma 4.1 there are most one τ 1 P Statept 1 q satisfying σW pt, τ q " W pt 1 , τ 1 q. Let us assume we are in this situation for a fixed pair pτ, τ 1 q. If f τ is trivial (is a forest with only trivial trees), then W pt, σq is a list of trivial words and thus so does W pt 1 , τ 1 q implying that f τ 1 is trivial. Therefore, α τ " α n´1 " α τ 1 where n is the number of leaves of t. Assume that f τ is non-trivial and consider a vertex ν of f τ that is connected to a leaf by an edge. Let rνs be the equivalence class of ν w.r.t. the relation " . Consider all geodesic paths c contained in f τ starting at a root and ending at a leaf that are passing through an element of rνs. Define the images c 1 of each of those paths inside f τ 1 as explained in Proof of (2) and put W the set of all last trivalent vertices (i.e. trivalent vertices connected to a leaf) of paths c 1 . It is easy to see that W is equal to an equivalence class rν 1 s for a certain vertex ν 1 of f τ 1 . The definition of the equivalence relation " implies that σ restricts to a bijection from the set of leaves that are descendant of vertices in the class rνs to the set of leaves that are descendant of vertices in the class rν 1 s. The order of the class rνs is equal to the number of leaves that are children of vertices in rνs divided by two and thus rνs and rν 1 s have same order. By (1), we have that the states τ and τ 1 take a unique value (0 or 1) for any element of rνs and rν 1 s that is τ pνq " τ 1 pν 1 q. Consider the forestsf ,f 1 that are the subforests of f τ , f τ 1 obtained by removing the set of vertices rνs, rν 1 s and edges starting from them, respectively. By applying our process tof ,f 1 we are able to show that αpf τ , τ q " αpf τ 1 , τ 1 q where αpf τ , τ q " α A β B for A (resp. B) the number of vertices of f τ for which τ takes the value 0 (resp. 1). The forest f τ and f τ 1 have necessarily the same number of vertices and thus so does z τ and z τ 1 . Since α τ " αpf τ , τ qα N where N is the number of vertices of z, we obtain that α τ " α τ 1 . 4.6. Splitting the sum over rooted subtrees. We further decompose the sum Φptqδ e " ÿ τ PStateptq α τ δ W pt,τ q by using rooted subtrees of t. Let Eptq be the set of all rooted subtrees of t (including the trivial one and t). For any z P Eptq we write Statept, zq for the set of states τ satisfying z τ " z, see Notation 4.2. We obtain the following decomposition: (4.5) Φptqδ e " ÿ zPEptq ÿ τ PStatept,zq α τ δ W pt,τ q .
Given z P Eptq we consider the unique forest f " f z satisfying that t " f z˝z . Fix a state τ P Statept, zq. For any trivalent vertex ν of z we have that τ pνq " 0 and there are npzq´1 of them if npzq denotes the number of leaves of z. If a leaf ν of z is a trivalent vertex of t (i.e. is not a leaf of t), then necessarily τ pνq " 1 by maximality of z " z τ . Let bpzq be the number of those. Then τ can take any values on the other vertices of t, that are the vertices of f that are not leaves of z (trivalent vertices of f that are not roots of f ). Note that there are nptq´npzq´bpzq such vertices and we set mpzq this number and V 1 pf q those trivalent vertices. We obtain the formula: where α 1,τ pf q is a monomial in α, β of degree mpzq that only depends on the restriction We obtain that npzq " 2, nptq " 5, bpzq " 1, mpzq " 2 and V 1 pf z q " tν 3 , ν 4 u. If τ P Statept, zq, then necessarily τ pν 1 q " 0, τ pν 2 q " 1 and τ can take any values at ν 3 and ν 4 . Equality (4.5) becomes (4.6) Φptqδ e " ÿ zPEptq α npzq´1 β bpzq ÿ τ PStatept,zq α 1,τ pf z qδ W pt,τ q .
Notation 4.5. Write Statept, zq`for the set of states τ satisfying that z τ " z and such that there exists τ 1 P Statept 1 q for which σW pt, τ q " W pt 1 , τ 1 q.
The following lemma provides a useful bound on the second part of the sum (4.7).
Lemma 4.6. If v " σ˝t s is a reduced fraction, then for any z P Eptq, we have that Proof. Fix z P Eptq and τ P Statept, zq`. Let f " f z be the unique forest satisfying that t " f˝z. It is easy to see that if ν P V 1 pf q and ν "ν withν P Vptq, then necessarilỹ ν belongs to V 1 pf q. We partition V 1 pf q as a union of equivalence classes rν 1 s,¨¨¨, rν k s w.r.t. the relation " where ν 1 ,¨¨¨, ν k is a set of representatives. Let m j be the number of elements in the class rν j s and note that mpzq " ř k j"1 m k . We obtain that α 1,τ pf z q " α m 1 τ,1¨¨¨α A state τ P Statept, zq`is thus completely characterized by its values at ν 1 ,¨¨¨, ν k . There are at most 2 k such states. Hence we obtain ÿ τ PStatept,zq`α where κ runs over all maps from t1,¨¨¨, ku to tα 2 , β 2 u. This sum is then equal to ś k j"1 ppα 2 q m j`p β 2 q m j q and thus (4.9) ÿ τ PStatept,zq`α 1,τ pf z q 2 ď k ź j"1 ppα 2 q m j`p β 2 q m j q.
This term tends to zero as n goes to infinity. So we only need to consider the rest of rooted subtrees for which mpzq ď hpnq.
Lemma 4.7. We have the inequality Proof. We start by proving that there exists a subset of vertices A Ă Vptq having hpnq elements that is contained in the vertex set of any rooted subtree z P Eptq satisfying that mpzq ď hpnq, i.e.
Recall that t is a tree with n leaves and thus has n´1 trivalent vertices. Consider the longest geodesic path c inside t starting from the root and ending at one leaf. We claim that the length |c| of this path is larger than 2hpnq`1. Assume by contradiction that any path in t has length less than 2hpnq. This implies that t is a rooted subtree of the full rooted binary tree having 2 2hpnq leaves all at distance 2hpnq from the root. This tree has 2 2hpnq´1 vertices that is 2 log 2 pn{2q´1 " n{2´1. Since t has n´1 vertices we obtain a contradiction. Therefore, there exists a path c P Pathptq of length larger than 2hpnq`1. The path c contains at least 2hpnq trivalent vertices of t. Consider a rooted subtree z P Eptq such that mpzq ď hpnq. There are at most hpnq`1 vertices of c that are not inside z. Those vertices are necessarily the one at the end of c that are the hpnq`1 last one. Therefore, Vpzq contains at least the hpnq first vertices of c. This proves that there is a subset A Ă Vptq of hpnq elements contained in every rooted subtree z P Eptq for which mpzq ď hpnq. Therefore, if τ is a state on t satisfying that mpz τ q ď hpnq, then τ pνq " 0 for any ν P A. Therefore, ÿ τ PStateptq mpzτ qďhpnq where γ runs over every maps from VptqzA Ñ t0, 1u and where α γ " α |γ´1p0q| β |γ´1p1q| . But ř γ α 2 γ " 1 and thus ÿ τ PStateptq mpzτ qďhpnq α 2 τ ď α 2hpnq .

4.7.
End of the proof. For v " σ˝t s a reduced fraction with trees having n leaves we have the following: φpvq " ÿ zPEptq α 2npzq´2 β 2bpzq ÿ τ PStatept,zq`α 1,τ pf z q 2 by (4.7) ď ÿ zPEptq α 2npzq´2 β 2bpzq pα 4`β4 q mpzq 2 by Lemma 4.6 ď ÿ zPEptq mpzqąhpnq Since lim nÑ8 hpnq " 8 and 0 ă α, α 4`β4 ă 1, we obtain that lim nÑ8 sup V zVn |φpvq| " 0 where V n is the subset of V of elements that can be written as a fraction of symmetric trees with less than n´1 leaves. Since pV n q n is an increasing sequence of finite subsets of V whose union is equal to V we obtain that φ vanishes at infinity.
Remark 4.8. We have proven that for any 0 ă α ă 1 the map φ α : V Ñ C is a positive definite function that vanishes at infinity. Moreover, lim αÑ1 φ α pvq " 1 for any v P V implying that V has the Haagerup property. This theorem was first proved by Farley where he defined a proper cocycle on V with value in a Hilbert space [Far03]. Using Schoenberg Theorem applied to the square of the norm of this cocycle we obtain a one parameter family of positive definite maps f α : V Ñ C, 0 ă α ă 1 satisfying that f α pvq " α 2npvq´2 where npvq is the minimum number of leaves for which v is described by a fraction of symmetric trees with npvq leaves. In [BJ19b], Jones and the author constructed a family of positive definite maps on V that coincide with the maps of Farley when restricted to Thompson's group T , see [BJ19b, Remark 1], but do not vanishes at infinity on the group V . A similar observation shows that the restriction to T of our maps φ α coincide with the maps of Farley. However, those three families of maps no longer coincide on the whole group V .

A class of wreath products with the Haagerup property
Following the preliminary section we consider a group Γ, an injective morphism S : Γ Ñ Γ ' Γ, the associated monoidal functor Ξ : F Ñ Gr, Ξp1q " Γ, ΞpY q " S, and the associated category C Ξ " C. Write G C for the group of fractions of the category C (at the object 1).

Constructions of unitary representations.
Given a representation of Γ and an isometry R : H Ñ H b H we want to construct a representation of the larger group G C . To do this we will define a monoidal functor Ψ : C Ξ Ñ Hilb and then use Jones' technology. We start by explaining how to build such a functor.
Proof. Consider a monoidal functor Ψ and the associated couple pρ, Rq. The two first properties come from the fact that morphisms of Hilb are linear isometries. The third property results from the computation of ΨpY˝gq and the equality Y˝g " Spgq˝Y inside the category C for all g P Γ. Since any morphism of C Ξ is the composition of tensor products of g P Γ, the tree Y , and some permutations we have that those properties completely characterized Ψ and are sufficient.
Note that a functor Ψ as above satisfies the equality Ψpf q˝ρ bn pgq " ρ bm pΞpf qpgqq˝Ψpf q, @f P F pn, mq, g P Γ n .
Assumption. From now one we assume that Spgq " pg, eq and thus the group of fractions G C is isomorphic to ' Q 2 Γ¸V by Proposition 2.10. We will build specific coefficients for G C using Jones' representations arising from Proposition 5.1.

5.2.
Constructions of matrix coefficients. From any coefficient of Γ and coefficient φ α of V (as constructed in Section 4.1) we build a coefficient of the larger group G C » ' Q 2 Γ¸V . Positive definite maps on the group Γ. Let φ Γ : Γ Ñ C be a positive definite function on Γ. There exists a unitary representation pκ 0 , K 0 q and a unit vector ξ P K 0 such that φ Γ pgq " xξ, κ 0 pgqξy for any g P Γ.
For technical purpose we consider the infinite tensor product of the representation κ 0 . In order to take an infinite tensor product we must first add a vector on which the group acts trivially. Define K :" K 0 ' CΩ where Ω is a unit vector and extend the unitary representation κ 0 as follows: κpgqpη ' µΩq " pκ 0 pgqηq ' µΩ for any g P Γ, η P K 0 , µ P C.
Hence, κ is the direct sum of κ 0 and the trivial representation 1 Γ . Let K 8 be the infinite tensor product b kě1 pK, Ωq with base vector Ω. In other words K 8 is the completion of the directed system of Hilbert spaces pK bn , n ě 1q with inclusion maps ι n`p n : K bn Ñ K bn`p , η Þ Ñ η b Ω bp for n, p ě 1.
For any g P Γ we define the following map: κ 8 pgqpb kě1 η k q " b kě1 κpgqη k for an elementary tensor b kě1 η k such that η k " Ω for k large enough. This formula defines for any n a unitary representation of Γ on K bn . This family of representations is compatible with the directed system of Hilbert spaces and thus defines a unitary representation Isometries for the Thompson group V . Consider 0 ď α ď 1 and the map R α : H Ñ H b H defines in Section 4.1. Hence, H " ℓ 2 pMq where M is the free monoid in four generators a, b, c, d. Moreover, recall that we write β for ? α 2´1 and we have Mixing representations of Γ with isometries. We can now build a monoidal functor from C to Hilb and a matrix coefficient for its group of fractions G C . Define the Hilbert space L :" K 8 b ℓ 2 pMq and the map: R " R φ Γ ,α : L Ñ L b L as follows: Note that up to flipping tensors we have the formula Rpη b δ x q " pη b ξ b|x|`1 q b R α pδ x q for x P M, η P K 8 .
Observe that in the formula we have ξ elevated to certain tensor powers. This will permit to have matrix coefficients tending quickly to 0 at infinity. This is the reason why we consider K 8 rather than K 0 . Define the unitary representation ρ :" κ 8 b 1 : Γ Ñ UpLq such that ρpgqpη b ζq " κ 8 pgqpηq b ζ for any g P Γ, η P K 8 , ζ P ℓ 2 pMq. The following proposition is straightforward: Proposition 5.2. The pair pρ, Rq verifies the assumptions of Proposition 5.1. Hence, there exists a unique monoidal functor Ψ " Ψ φ Γ ,α : C Ξ Ñ Hilb satisfying that Ψp1q " L, ΨpY q " R φ Γ ,α and Ψpgq " ρpgq for any g P Γ.
Let us apply the Jones construction to the functor Ψ " Ψ φ Γ ,α of the proposition. We obtain a Hilbert space L φ Γ ,α and a unitary representation of the group of fractions of C " C Ξ that is: π φ Γ ,α : G C Ñ UpL φ Γ ,α q. We now build a coefficient for G C . Consider the unit vector ξ b δ e P L view as a vector of the larger Hilbert space L " L φ Γ ,α and set ϕ φ Γ ,α : G C Ñ C, v g Þ Ñ xπ φ Γ ,α pv g qξ b δ e , ξ b δ e y.
Lemma 5.3. Let t be a tree and τ a state on t. Decompose t as f τ˝zτ (see Notation 4.2). Consider the geodesic path in f τ starting at a root and ending at the j-th leaf and its subpath with same start but ending at the last right-edge of the path. If this subpath is empty (has length zero), we set L j pτ, tq " L j pτ q " 1. Otherwise, we set L j pτ, tq " L j pτ q the length of this subpath. We have that (up to the identification L bn » pK 8 q bn b ℓ 2 pM n q) where ξ bLpτ q :" ξ bL 1 pτ q b¨¨¨b ξ bLnpτ q P pK 8 q bn and where W pt, τ q is the list of words in the free monoid M defined in Section 4.3.
The proof follows from an easy induction on the number of vertices of f τ . Rather than proving it we illustrate the formula on one example. Consider the following tree: Define the state τ such that τ pν 1 q " 0, τ pν 2 q " 1, τ pν 3 q " 0, τ pν 4 q " 1. We then have that z τ " Y and f τ " t 2 b I where t 2 is the full rooted binary tree with 4 leaves all at distance 2 from the root. Since τ takes the value 0 twice and the value 1 twice we obtain that α τ " α 2 β 2 . Following each geodesic path from the root to the j-th leaf and considering the state τ at each vertex we obtain that W pt, τ q " pca, cb, dc, dd, eq.
The geodesic path in f τ from a root to the first leaf is a succession of two left-edges. So the subpath ending with a right-edge is trivial and thus has length zero. We then put L 1 pτ q " 1. The second subpath is a left-edge followed by a right-edge and thus L 2 pτ q " 2. Looking at the other leaves we obtain that L 1 pτ q " 1, L 2 pτ q " 2, L 3 pτ q " 1, L 4 pτ q " 2, L 5 pτ q " 1. Applying the formula of the proposition we get that the τ -component of Φptqpξ b δ e q is equal to Another way to compute L j pτ q is to look at the longest subword of W pt, τ q j starting at the first letter and ending at the last b or d-letter. If this words is trivial (there are no b or d-letter) we put L j pτ q " 1. Otherwise, L j pτ q is the length of this word.

5.3.
Matrix coefficients vanishing at infinity and the Haagerup property. The next proposition proves that a large class of matrix coefficients of G C vanish at infinity. This is the key technical result for proving that wreath products have the Haagerup property.
Proposition 5.4. Consider a discrete group Γ and a positive definite map φ Γ : Γ Ñ C satisfying that there exists 0 ď c ă 1 such that |φ Γ pgq| ď c for any g ‰ e and that vanishes at infinity. If 0 ă α ă 1 and ϕ " ϕ φ Γ ,α is the coefficient built in Section 5.2, then it vanishes at infinity.
Proof. Consider trees t, t 1 with n leaves, a permutation σ P S n and g " pg 1 ,¨¨¨, g n q P Γ n .
Write v " σt t 1 P V and v g " gσt t 1 P G C . Recall that any element of G C can be written in that way. Fix 0 ă ε ă 1 and assume that |ϕpv g q| ě ε. Let us show that there are only finitely many such v g . By definition of the coefficients we have that |ϕpv g q| ď ś n j"1 |φ Γ pg j q|. Since the map φ Γ : Γ Ñ C vanishes at infinity and |ϕpv g q| ě ε we obtain that there exists a finite subset Z Ă Γ such that g P Z n .
Observe that |ϕpv g q| ď |φ α p σt t 1 q| where φ α : V Ñ C is the coefficient built in Section 4.1. We proved in Section 4 that φ α vanishes at infinity. Therefore, we may write σt t 1 as a fraction with few leaves. Hence, there exists a fixed N ě 1 depending solely on ε such that θt N s for some tree s and permutation θ and where t N denotes the full rooted binary tree with 2 N leaves all at distance N from the root The next claim will show that the fraction gσt t 1 can be reduced as a fraction g 1 θt N 1 s for some N 1 ě 1 that only depends on N (and thus only depends on ε). To do this we need to show that if g j is nontrivial, then the geodesic path inside t ending at the j-th leaf is mainly a long path with only left-edges. Define P j to be the geodesic path from the root of the tree t to the j-th leaf of t and write P R j its subpath starting at the root and ending at the last right-edge of P j . Claim: We have the inequality (5.2) |ϕpv g q| ď p|P R j |`1q maxpα 2 , |φpg j q|q |P R j | for any 1 ď j ď n. Proof of the claim: Lemma 5.3 states that Therefore, By Propostion 4.4, we have that given a state τ P Stateptq there are at most one τ 1 P Statept 1 q such that W pt 1 , τ 1 q " σW pt, τ q and in that case α τ " α τ 1 . This implies that Fix 1 ď j ď n and consider the set of vertices of the path P R j that we denote from bottom to top by ν 1 , ν 2 ,¨¨¨, ν q . Our convention is that the last vertex ν q is the source of the last edge of P R j and thus |P R j | " q. Define tτ P Stateptq : τ pν 1 q " 1u if k " 0; tτ P Stateptq : τ pν 1 q "¨¨¨" τ pν k q " 0, τ pν k`1 q " 1u if 1 ď k ď q´1; tτ P Stateptq : τ pν 1 q "¨¨¨" τ pν q q " 0u if k " q.
Observe that ÿ Moreover, if τ P S k for 0 ď k ď q, then L j pτ q " q´k. Therefore, This proves the claim.
We now explain how to reduce our fraction gσt s .
Claim: There exists Q ě 1 such that |P R j | ď Q for any j P J where J :" tj : g j ‰ eu is the support of g. If J is empty, then we can take Q " 1. Assume J is nonempty and take j P J. By assumption we have that |φpg j q| ă c for a fixed constant 0 ă c ă 1. Moreover, 0 ă α ă 1. This implies that the quantity pP`1q maxpα 2 , |φpg j q|q P tends to zero in P . Therefore, by the preceding claim we deduce that there exists Q ě 1 such that |P R j | ď Q for any j P J. This proves the claim. From the claim we deduce that the geodesic path P j from the root of t to its j-th leaf with j P J is the concatenation of a first path P R j of length less than Q ending with a right-edge and a second path which consists on a succession of left-edges. Using the rules of composition of morphisms in the category C Ξ we can write the composition g˝σ˝t in a different fashion as follows. First observe that g˝σ " σ˝g σ where g σ P Γ n whose i-th component is g σpiq . Second we make the group elements go down in the tree using the relation px, eqY " Y x for x P Γ. We apply this relation to any nontrivial group element g j , j P J along the second part of the path P j that is a succession of left-edges. We obtain that g˝σ˝t " f˝σ 1˝g1˝t1 for some f, σ 1 , g 1 , t 1 satisfying that σt " f σ 1 t 1 and such that g 1 P Z n 1 for some n 1 ď n. We can choose t 1 for which every leaf is at most at distance Q from the root and thus can be seen as rooted subtree of the complete binary tree t Q that has 2 Q leaves all of them at distance Q from the root. We obtain that v g " f σ 1 g 1 t Q f 1 t 2 .
Using (5.1), we obtain that v g can be reduced as a fraction g 1 σ 1 t U t 2 where U " maxpN, Qq and g 1 P Z n 1 where n 1 " 2 U . Since Z is finite and U is fixed (and only depends on ε) there are only finitely many such fractions implying that ϕ vanishes at infinity.
We are now able to prove one of the main theorems of this article.
Theorem 5.5. If Γ is a discrete group with the Haagerup property, then so does the wreath product ' Q 2 Γ¸V .
Proof. Fix a discrete group Γ with the Haagerup property. By Proposition 2.10 the wreath product ' Q 2 Γ¸V is isomorphic to the group of fractions G C and thus it is sufficient to prove that this later group has the Haagerup property. Consider a finite subset X Ă G C and 0 ă ε ă 1. Since X is finite there exists n and a finite subset Z Ă Γ such that X Ă X n where X n is the set of fractions v g :" gσt s where t, s are trees with n leaves, g " pg 1 ,¨¨¨, g n q P Z n and σ P S n . Fix ε 1 ą 0 the unique positive number satisfying that p1´ε 1 q 2n`n 2 " 1´ε.
Since Γ has the Haagerup property there exists a positive definite map φ Γ : Γ Ñ C vanishing at infinity satisfying that |φ Γ pxq| ą 1´ε 1 for any x P Z.
Hence, for any finite subset X Ă G C and 0 ă ε ă 1 there exists a positive definite map ϕ : G C Ñ C vanishing at infinity and satisfying that |ϕpvq| ě 1´ε for any v P X. This implies that G C has the Haagerup property.

5.4.
Haagerup property for twisted wreath products. In this section we fix a group Γ and an automorphism of it θ P AutpΓq. Recall from Section 2.3.2 that this defines a category C " C Γ,θ where morphisms are forests with leaves labelled by elements of Γ and by permutations satisfying the relation Y˝g " pθpgq, eq˝Y.
Moreover, the group of fractions of C is isomorphic to the twisted wreath product Γ ≀ θ Q 2 V. By adapting the proof of Theorem 5.5 we obtain the following result.
Theorem 5.6. If Γ is a group with the Haagerup property and θ P AutpΓq is an automorphism, then the twisted wreath product Γ ≀ θ Q 2 V has the Haagerup property. Proof. Fix a group Γ with the Haagerup property and an automorphism θ of it. Denote by G the twisted wreath product Γ ≀ θ Q 2 V that we identify with the group of fractions of the category C Γ,θ . We mainly follow the construction explained in Section 5.2 and keep similar notations. We choose a positive definite function φ Γ : Γ Ñ C realized as φ Γ pgq " xξ, κ 0 pgqξy and put K " K 0 ' CΩ. Consider K 8 :" b ně0 pK, Ωq and the associated representation of Γ denoted by κ 8 . Now, we modify the construction by considering the automorphism θ. We define K θ :" ' nPZ K 8 the infinite direct Hilbert space sum of K 8 over the set Z and the representation κ θ :" ' nPZ pκ 8˝θ´n q.
We set L :" K θ b ℓ 2 pMq and the unitary representation ρ θ :" κ θ b 1 similarly than before. We now define our R-map. To do this we need to replace our favourite vector ξ by one that is almost invariant by the shift operator. Given any vector η P K and n ě 1 we put: 2n`1 ' kPZ η χ r´n,ns pkq P K θ where χ r´n,ns is the characteristic function of tk P Z : |k| ď nu. Note that if η is a unit vector, then η n is again a unit vector satisfying xshiftpη n q, η n y " 2n 2n`1 . We will then consider vectors like ξ n and ξ b|x|`1 n in K θ . The new R-map from L to LbL is the following: for x ‰ e. It is the same formula than in the untwisted case except that η, ξ, ξ b|x|`1 are replaced by shiftpηq, ξ n , ξ b|x|`1 n , respectively. By reordering the tensors we obtain the following short formula: One can check that pR, ρq defines a monoidal functor from F to Hilb and thus a Jones' representation π : G Ñ UpL q. We consider the positive definite function: ϕ :" ϕ n,α,φ Γ pγq :" xπpγqξ n b δ e , ξ n b δ e y for any γ P G.
A similar proof can be applied by considering φ Γ as in the proof of Theorem 5.5, letting α tending to one and n to infinity. We then obtain a net of positive definite functions ϕ n,α,φ Γ vanishing at infinity and tending to one thus proving that the group of fraction G has the Haagerup property.
The following proposition shows that we have many new examples of wreath products with the Haagerup property; indeed the wreath product Γ ≀ θ Q 2 W with W being F, T, or V remembers the group Γ and the automorphism θ. It was proven in [Bro22a,Theorem 4.12] for the V -case. The untwisted version of it has been proven for the F and T -cases in [Bro22b,Theorem 4.1] and can easily be extended to the twisted case. We leave the proof of this extension to the reader.
Proposition 5.7. Consider two pairs of groups with an automorphism pΓ, θq and pΓ,θq. Let G,G be the associated twisted wreath products Γ ≀ θ Q 2 V andΓ ≀θ Q 2 V . We have that G »G if and only if there exists an isomorphism β : Γ ÑΓ and h PΓ satisfying θ " adphq˝βθβ´1. The same result holds when V is replaced by F or T .

Groupoid approach and generalisation of the main result
In this section we adopt a groupoid approach. We include all necessary definitions and constructions that are small modifications of the group case previously explained in the preliminary section. This leads to proofs of Theorem C and Corollary D.
6.1. Universal groupoids. We refer to [GZ67] for the general theory on groupoids and groups of fractions.
Definition 6.1. A small category C admits a calculus of left-fractions if: ‚ (left-Ore's condition) For any pair of morphisms p, q with same source there exists some morphisms r, s satisfying rp " sq; ‚ (Weak right-cancellative) If pf " qf , then there exists g such that gp " gq.
To any category C can be associated a universal (or sometime called enveloping) groupoid pG C , P q together with a functor P : C Ñ G C . The groupoid G C has the same collection of objects than C and morphisms are signed paths inside the category C: compositions of morphisms of C and their formal inverse. The next proposition shows that if C admits a calculus of left-fractions then any morphism of G C can be written as P ptq´1P psq for some morphisms of t, s of C with same target and thus justifies the terminology. The proof can be found in [GZ67, Chapter I.2].
Proposition 6.2. If C admits a calculus of left-fractions, then any morphism of G C can be written as P ptq´1P psq for t, s morphisms of C (having common target). Using the fraction notation t s :" P ptq´1P psq we obtain that f t f s " t s for any morphism f of C. Moreover, we have the following identities: t s t 1 s 1 " f t f 1 s 1 for any f, f 1 satisfying f s " f 1 t 1 ; andˆt s˙´1 " s t .
We say that G C is the groupoid of fractions of C.
where ξ a , η a are the components of ξ, η in H a . Given a fraction f f 1 with f P Cpa, bq, f 1 P Cpa 1 , b 1 q we define a partial isometry πˆf f 1˙o nH with domain H a 1 and range H a We say that pπ,L q is a representation of the groupoid G C .
6.3. Important examples. Higman-Thompson's groups. If we consider SF k the category of k-ary symmetric forests, then it is a category that admits a calculus of leftfractions for k ě 2. Note that SF 2 " SF is the category of binary symmetric forests which we worked with all along this article. Observe that the group of automorphisms G SF k pr, rq can be represented by pairs of symmetric k-ary forests with both r ě 1 roots and the same number of leaves. This is one classical description given in the article of Brown of the so-called Higman-Thompson's group V k,r [Hig74,Bro87]. Hence, the groupoid G SF k contains (in the sense of morphisms) every Higman-Thompson's group V k,r for a fixed k ě 2. Larger categories. We consider larger categories made of symmetric forests and groups. Fix k ě 2 and consider a group Γ together with a morphism θ : Γ Ñ Γ. Define the morphism S k : Γ Ñ Γ k , g Þ Ñ pθpgq, e,¨¨¨, eq. We can now proceed as in Section 2.3.1 for constructing a monoidal functor Θ : SF k Ñ Gr and a larger category Cpk, θ, Γq. The only difference being that morphisms of SF k are all composition of tensor products of the trivial tree I and the unique k-ary tree Y k (instead of the binary tree Y ) that has k leaves. We then set Θp1q " Γ, ΘpY k q " S k and the definition of the larger category Cpk, θ, Γq becomes obvious. It is a category that admits a calculus of left-fractions. By adapting Proposition 2.10 we obtain the following: Proposition 6.5. Consider k ě 2 and the identity automorphism θ " id. Let C k be the category Cpk, id, Γq and put G k its universal groupoid. If r ě 1, then the automorphism group G k pr, rq of the object r is isomorphic to the wreath product Γ ≀ Qrp0,rq V k,r :" ' Q k p0,rq Γ¸V k,r for the classical action of the Higman-Thompson's group V k,r on the set Q k p0, rq of k-adic rationals in r0, rq. More generally, if θ is any automorphism of Γ, then G k pr, rq is isomorphic to the twisted wreath product Γ ≀ θ Qrp0,rq V k,r :" ' Q k p0,rq Γ¸θ V k,r where the action V k,r ñ ' Q k p0,rq Γ is the following: pv¨aqpxq :" θ log k pv 1 pv´1xqq papv´1xqq for v P V k,r , a P ' Q k p0,rq Γ, x P Q k p0, rq.
Remark 6.6. Note that given a fixed k ě 2, we have that two objects r 1 , r 2 of the universal groupoid G SF k are in the same connected component if and only if r 1 " r 2 modulo k´1. In that case the automorphism groups of the objects r 1 and r 2 inside G SF k are isomorphic (to see this: simply conjugate the first automorphism group by any morphism f P G SF k pr 1 , r 2 q) and thus V k,r 1 » V k,r 2 . The same argument applies to the wreath products associated to C k :" Cpk, θ, Γq. This provides isomorphisms between various wreath products of the form Γ ≀ θ Q k p0,rq V k,r . In particular, if k " 2, then all Higman-Thompson's groups V 2,r (and wreath products Γ≀ θ Q 2 p0,rq V 2,r for fixed pΓ, θq) are mutually isomorphic but this is no longer the case when k is strictly larger than 2.
6.4. Haagerup property for groupoids. Haagerup property was defined for measured discrete groupoids by Anantharaman-Delaroche in [AD12]. Her work generalises two important cases that are countable discrete groups and measured discrete equivalence relations. Our case is slightly different as fibers might not be countable. However, since the set of objects is countable we can study our groupoid in a similar way than a discrete group and avoid any measure theoretical considerations. Let G be a small groupoid with countably many objects. We recall what are representations and coefficients for G. Identify G with the collection of all morphisms of G. A representation pπ, L q of G is a Hilbert space L equal to a direct sum ' aPobpGq L a and a map π : G Ñ BpL q such that πpgq is a partial isometry with domain L sourcepgq and range L targetpgq . A coefficient of G is a map φ : G Ñ C, g Þ Ñ xη, πpgqξy for a representation pπ, L q and some unit vectors ξ, η P L . The coefficient is positive definite (or is called a positive definite function) if η " ξ. Note that equivalent characterizations of positive definite functions exist in this context but we will not need them. We define the Haagerup property as follows.
Definition 6.7. A small groupoid G with countably many objects has the Haagerup property if there exists a net of positive definite functions on G that converges pointwise to one and vanish at infinity.
Assume that G has countable fibers and is as above. Let µ be any strictly positive probability measure on the set of objects of G. Then we can equip pG, µq with a structure of a discrete measured groupoids, see [AD12]. The two notions of coefficients and positive definite functions coincide for G and pG, µq. Moreover, G has the Haagerup property in our sense if and only if pG, µq does in the sense of Anantharaman-Delaroche [AD12] which justifies our definitions. The following property is obvious.
Proposition 6.8. Let G be a small groupoid with countably many objects. Consider a subgroupoid G 0 in the sense that obpG 0 q Ă obpGq and G 0 pa, bq Ă Gpa, bq for any objects a, b of G 0 . If G has the Haagerup property, then so does G 0 and in particular every group Gpa, aq (considered as a discrete group) for a P obpGq.
Proof of Theorem C and Corollary D. Consider a discrete group Γ with the Haagerup property and an injective morphism θ : Γ Ñ Γ. This defines a map S k : Γ Ñ Γ k , a category C " Cpk, θ, Γq with universal groupoid G C as explained above. Note that G C is a small category with set of object N˚that is countable. Let us prove that G C has the Haagerup property. We prove the case k " 2. The general case can be proved in a similar way. Claim: We can assume that θ is an automorphism. This follows from [Bro22a, Section 4.1]. Indeed, from pΓ, θq we construct a directed system of groups indexed by the natural numbers where all groups are Γ and the connecting maps are θ. The limit is a group p Γ that admits an automorphism p θ. Now, if Γ has the Haagerup property, then so does p Γ since it is the limit of a group with the Haagerup property. Note, this fact uses crucially that θ is injective (and thus no quotients are performed). Moreover, we prove in Proposition 4.3 of [Bro22a] that the groupoid of fractions G C of Cp2, θ, Γq is isomorphic to the groupoid of fractions of the category Cp2, p θ, p Γq.
From now one we assume that θ is an automorphism. Consider a pair pρ, Rq constructed from a positive definite coefficient φ Γ : Γ Ñ C vanishing at infinity and an isometry R α for some 0 ă α ă 1 as in Section 5.2. Assume that there exists 0 ď c ă 1 such that |φ Γ pgq| ă c for any g ‰ e. This defines a functor Ψ : C Ñ Hilb that provides a representation pπ,L q of the universal groupoid G C satisfying that πˆg σf f 1˙f 1 ξ " pf Tenspσ´1qρ bn pg´1qΨpqqξ for f, f 1 forests with n leaves, σ P S n and g P Γ n . Consider the unit vector η N,φ Γ ,α :" N´1 {2 ' N n"1 ξ b δ e for N ě 1 and where ξ is the vector satisfying φ Γ pgq " xξ, κ 0 pgqξy, see Section 5.2. By following the same proof than Proposition 5.4 we obtain that the coefficient ϕ N,φ Γ ,α associated to η N,φ Γ ,α and pπ,L q vanishes at infinity. Fix a net of positive definite functions pφ i : Γ Ñ C, i P Iq satisfying the hypothesis of the Haagerup property such that |φ i pgq| ă c i for any g ‰ e for some 0 ď c i ă 1. The net of coefficients pϕ N,φ i ,α , N ě 1, i P I, 0 ă α ă 1q on the groupoid G C satisfies all the hypothesis required by the Haagerup property. This proves Theorem C. Consider the category C " Cp2, θ, Γq where Γ has the Haagerup property, θ P AutpΓq is an automorphism, and the category of k-ary forests SF k . By Proposition 6.8 we have that the group G C pr, rq of automorphisms of the object r in the universal groupoid of C is isomorphic to the twisted wreath product Γ ≀ θ Q k p0,rq V k,r . We proved that G C has the Haagerup property and thus so does the isotropy group G C pr, rq (by Proposition 6.8). This proves Corollary D.

Appendix A. Categories and groups of fractions
We end this article by providing an alternative description of Jones' actions using a more categorical language. We do not give details and only sketch the main steps. This was explained to us by Sergei Ivanov, Richard Garner and Steve Lack. We are very grateful to them. We keep the notation of Section 2.2 and thus Φ : C Ñ D provides a Jones' action π Φ : G C ñ X with X " lim Ý Ñt,Φ X t . Let pG C , P q be the universal groupoid of C with functor P : C Ñ G C . Let pe Ó Cq be the comma-category of objects under e whose objects are morphisms of C with source e and morphism triangles of morphisms of C (e.g. if C " F , e " 1, then objects and morphisms of p1 Ó F q are trees and forests respectively). This category comes with a functor pe Ó Cq Ñ C consisting in only remembering the target of morphisms (e.g. sending a tree to its number of leaves and keeping forests for morphisms). The composition of functorsΦ : pe Ó Cq Ñ C Ñ D provides a diagram of type pe Ó Cq in the category D and the colimit (if it exists) corresponds to our limit X . Assume that the left Kan extension Lan P pΦq : G C Ñ D of Φ along P exists. Then one can prove that Lan P pΦqpeq is isomorphic to the colimit ofΦ and is thus isomorphic to X . But then Lan P pΦq sends G C pe, eq » G C in the automorphism group of X which corresponds to the Jones' action π Φ . Using this construction, if we only want a map from the group of fractions G C to the automorphism group of an object, then we don't need to require that objects of D are sets. Actions of the whole universal groupoid G C can be constructed in a similar way. In order to make this machinery working we need to have a target category D with sufficiently many colimits in order to have a Kan extension of our functor.