A machinery developed by the second author produces a rich family of unitary representations of the Thompson groups F, T and V. We use it to give direct proofs of two previously known results. First, we exhibit a unitary representation of V that has an almost invariant vector but no nonzero -invariant vectors reproving and extending Reznikoff’s result that any intermediate subgroup between the commutator subgroup of F and V does not have Kazhdan’s property (T) (though Reznikoff proved it for subgroups of T). Second, we construct a one parameter family interpolating between the trivial and the left regular representations of V. We exhibit a net of coefficients for those representations which vanish at infinity on T and converge to 1 thus reproving that T has the Haagerup property after Farley who further proved that V has this property.
Let be the usual Thompson groups acting on the unit interval . In , it was shown that certain categories with a privileged object give rise to a group of fractions and that a functor provides an action of the group . Similar ideas in the context of semigroups were developed by Ore; see for instance . Thompson’s groups can be constructed in this way using various categories of forests written . If is the category of Hilbert spaces with isometries for morphisms, then the functor Φ gives us a unitary representation of . In this article, we consider functors such that and , where is a fixed isometry and is the forest with n trees all of which are trivial except the ith one that has two leaves. Since any tree is a composition of , we obtain a well defined functor and thus a unitary representation of F that we extend to V via permutations of the tensors; see Section 1 for more details.
In Section 2, we construct a unitary representation of V having an almost invariant vector but no nonzero -invariant vectors showing that any intermediate subgroup between and V does not have Kazhdan’s property (T) . Recall that T was proved to not have property (T) initially by Reznikoff . Another proof consists in embedding T in the group of diffeomorphisms of the circle via the smoothing methods of Ghys and Sergiescu  or Thurston (described in ) and using results of Navas concerning diffeomorphism groups that imply that T does not have property (T) .
In Section 3, we give a one parameter family (, ) of unitary representations of V interpolating the trivial and the left regular one. We define some positive definite maps from those representations for which we can explicitly compute their values on T. They actually coincide on T with the family of maps constructed from the proper cocycle of Farley (see Remark 1), but differ on the larger group V. Hence they provide a positive definite approximation of the identity for T, thus proving Farley’s result that T has the Haagerup property (though Farley actually proved it for all of V) [1, 4]. Note that a geometric approach to Haagerup and Kazhdan properties for Thompson’s groups follows from the actions on CAT(0) cube complexes constructed by Farley [3, 5] (which is related to Chatterji, Drutu and Haglund’s characterization of these properties via median metric spaces).
1 Definitions and notations
We briefly recall the construction of actions of groups of fractions for the particular cases of F, T and V and refer to [7, 2] for more details. Let be the category of (binary planar) forests whose objects are the natural numbers and morphisms the set of binary planar forests with n roots and m leaves. We think of them as planar diagrams in the plane whose roots and leaves are distinct points in and , respectively, and are counted from left to right. We compose forests by stacking them vertically so that is the forest obtained by stacking on top of q the forest p, where the ith root of p is attached to the ith leaf of q. We obtain a diagram in the strip that we rescale in . For any , , we consider the forest , or simply if the context is clear, the forest with n roots and leaves, where the ith tree of has two leaves. For example,
Consider the set of pairs of trees with the same number of leaves, that we quotient by the relation generated by for any forest p. We write for the equivalence class of . These form the group of fractions of the category with the multiplication and inverse . It is isomorphic to Thompson’s group F.
Now consider the category of symmetric forests with objects and morphisms , where is the symmetric group of m elements. Graphically, we interpret a morphism as the concatenation of two diagrams. On the bottom, we have the diagram explained above for the forest p in the strip . The diagram of τ is the union of m segments , , in , where the are m distinct points in such that is on the left of . The full diagram of is obtained by stacking the diagram of τ on top of the diagram of p such that is the ith leaf of p. Given symmetric forests , , let be the number of leaves of the ith tree of p. Then we define the composition of morphisms by , where is the forest obtained from p by permuting its trees such that the ith tree of is the th tree of p and is the permutation corresponding to the diagram obtained from τ, where the ith segment is replaced by parallel segments. Thompson’s group V is isomorphic to the group of fractions of the category . Hence any element of V is an equivalence class of a pair of symmetric trees. Consider and the standard dyadic partitions and of associated to the trees s and t, respectively. The element g acting on is the unique piecewise linear function with positive constant slope on each that maps onto for any .
Consider the cyclic group a subgroup of the symmetric group and the subcategory of affine forests, where . The group of fractions of is isomorphic to Thompson’s group T. We will often identify and as subcategories of giving embeddings at the group level .
We say that a pair of symmetric trees is reduced if there are no pairs such that has strictly less leaves than t and such that .
Let be the category of complex Hilbert spaces with isometries for morphisms. Given an isometry , we construct a functor
for . Consider the quotient space
where the equivalence relation is generated by for all .
This quotient space has a pre-Hilbert structure given by that we complete into a Hilbert space . Note that is the inductive limit of the system of Hilbert spaces for trees t such that the embedding is given by . We denote by or the equivalence class of inside and identify and as subspaces of . We have a unitary representation given by the formula that we extend to the group V as follows:
Note that if are in the small Hilbert space and , then
Consider an orthonormal basis of the Hilbert space . The isometry R can be thought of as a possibly infinite matrix with three indices such that . We can reinterpret for a forest f as a partition function. Given a forest f with n roots and m leaves, we define the set of states on f as maps ω from the edges of f to the set of indices I. A vertex of f is a trivalent vertex, and thus roots and leaves are not vertices. If ω is a state on f and v a vertex, then we put the scalar equal to , where is the edge with target v and are the edges with source v which goes to the left and right, respectively. Consider some multi-indices and , and say that a state is compatible with if and for all , , where is the edge with source the kth root of f and is the edge with target the th leaf of f.
Using the notations of above, we have
with the convention that a product (resp. a sum) over an empty set is equal to one (resp. zero).
The infinite sum converges since the scalars are matrix coefficients of an isometry. This formula can be proved by induction on the number of leaves and using the fact that any forest is the composition of some elementary forests . As an illustration, we compute explicitly this inner product for the simple case . Fix some multi-indices and . Observe that if is nonempty, then necessarily and . Moreover, a state that is compatible with is completely determined by the couple of multi-indices , so there is at most one term in the sum of the last lemma. Moreover, the forest f has only one vertex (roots and leaves are not considered as vertices), and so there is only one term in the product that is . We obtain . If the trees have more than one vertex, then the product formula becomes more complicated. For example, if we have the state
that we represent with the spin of an edge (i.e., the image by ω of this edge) next to it, then
2 Kazhdan’s property (T)
Recall that a countable discrete group G has Kazhdan’s property (T) if any unitary representation having an almost invariant vector has in fact a nonzero invariant vector .
If is a unitary and is a unit vector, then the map
is an isometry which provides us a unitary representation as described in Section 1. We claim that if , then π has no nonzero -invariant vectors. Consider the following four trees
and let be the complete binary trees with leaves. Put , and . Note that
Define the element
where is the forest with roots in which every tree is a copy of a. Similarly, define
and observe that . Therefore, is in the commutator subgroup . Observe that
and thus with .
If , are elementary tensors of , then
By linearity and density, we obtain that for any vectors . Assume that is an -invariant unit vector and that . This implies . By density, there exists n and a unit vector such that . We obtain
for n as large as we want, which implies a contradiction since and proves the claim.
Consider the shift operator and the characteristic function of divided by , where , . Let be the associated sequence of unitary representations of V, and put as their direct sum. Note that we have for any , and thus the claim implies that π does not have any nonzero -invariant vectors. Let be the elementary tensor of , where each entry is equal to that we identify with the fraction viewed as an element of . Define to be the unit vector of , which is equal to in the mth slot and zero elsewhere. We claim that the sequence is an almost V-invariant vector. Fix , and note that, for m big enough, there exists , such that . By increasing m, we can also assume that there exists such that . We obtain
Since , any branch of p (resp. q) has length smaller than or equal to (resp. m). This implies that any component of and is equal to for some . Since the map is decreasing, we obtain
and thus is an almost V-invariant vector. Since π does not have any nonzero -invariant vectors, we obtain that any intermediate group between and V does not have property (T).
3 Haagerup property
Recall that a countable discrete group G has the Haagerup property if there exists a sequence of positive definite functions which vanish at infinity on G and such that for all (see ).
Consider the free group freely generated by , and let be the classical orthonormal basis of the Hilbert space . Identify with and with . Set , and define for the isometry
This defines a functor and a unitary representation as described in Section 1. The associated infinite matrix
is particularly simple since
and zero elsewhere.
Consider the case when and thus for any and zero elsewhere. Let f be a forest with n roots and m leaves, and observe that
The ith component is the word in written from right to left corresponding to the path from a root of f to its ith leaf such that a left turn (resp. right turn) contributes in adding the letter “a” (resp. the letter “b”). For example, if , then
The next lemma proves that if t is a tree, then the set of words in the tuple remembers completely t.
Consider two trees with n leaves. Assume that there exists a permutation acting on the leaves such that . Then and .
We prove the lemma by induction on the number of leaves . The result is immediate for and is also clear for since there is only one tree with two leaves and thus . The permutation σ is necessarily trivial.
Suppose the result is true for any k between 1 and n, and consider trees with leaves and a permutation σ such that . Note that there exist trees such that and , where is the forest with two roots whose first tree is and whose second is . Note that the word finishes by the letter a (resp. the letter b) if and only if i is a leaf of (resp. a leaf of ). We have the same characterization for the leaves of t, and thus necessarily σ realizes a bijection from the leaves of onto the leaves of for . Observe that for any leaves of . This implies . Similarly, we have , and thus, by the induction hypothesis, we have , , and σ is the identity on the leaves of and on the leaves of implying and . ∎
The lemma implies that contains the left regular representation of V. Indeed, consider some symmetric trees in with the same number of leaves. We have
This is nonzero (and then equal to one) if and only if . In that case, Lemma 2 implies and , and thus is the trivial group element. We obtain that the cyclic representation generated by for is the left regular representation of V. If , then the cyclic representation generated by is the trivial one. Indeed, if t is a tree with n leaves, then with n tensors, and thus coefficient (3.1) is always equal to one for any choice of . Our family of representations of V provides an interpolation between the trivial and the left regular representations.
Consider the family of positive definite maps
We will show that they vanish at infinity for and tend to the identity when α tends to one. Fix , and write R instead of . Section 1 tells us that, given a tree t with n leaves, we have
for any multi-index . Fix , and assume that the ω coefficient of above is nonzero for a certain . Then there exists a maximal subrooted tree of t such that the spin (i.e., the value of ω) at each of its edges is the trivial group element . Any (trivalent) vertex v of the tree satisfies . If is the unique forest satisfying , then any root w of that is not a leaf of satisfies since spins around it are necessarily . Then any other vertex u of has its spins around it equal to for some , and thus . We obtain
where is the number of leaves of that are not leaves of t (i.e., the number of nontrivial trees of ) and corresponds to the number of (trivalent) vertices of the tree . Indeed, a binary tree with n leaves has bifurcations (i.e., trivalent vertices). The spin of the edge with target the th leaf of t is the word in corresponding to the path in the forest starting at the root connected to and finishing at the leaf . Write for this word and for the corresponding n-tuple, and note that, by definition, the multi-index is equal to . Therefore, given a multi-index , there is at most one state compatible with and having a nonzero coefficient . Moreover, has to be of the form for some subrooted tree . Conversely, any subrooted tree z of t provides a unique state that is compatible with defined inductively by
where is the set of subrooted trees (including the trivial subtree) of t.
Consider a pair of symmetric trees and . We have
Fix a reduced pair of affine trees and the group element that is in Thompson’s group T since our trees are affine. We will show that all the terms in the sum (3.2) are equal to zero but one.
Consider some forests both of them having m leaves, and let be a cyclic rotation. If , then p and q have the same number of roots n, and there exists such that the jth tree of p is equal to the th modulo n tree of q for any .
Observe that if f is a forest, then the word is a power of a (resp. a power of b) if and only if i corresponds to the first leaf (resp. the last leaf) of a tree of f. Consider as above, and write and for the jth and kth trees of p and q, respectively. Fix j, and note that the observation implies that there exist some natural numbers such that the first and the last leaves of are sent to the first leaf of and the last leaf of . If modulo the number of roots of q, then would be equal to a tuple of a forest f having at least two trees and thus having at least two words that are powers of b, which is impossible. Therefore, σ realizes a bijection from the leaves of onto the leaves of for a certain c. Since σ is cyclic, the number c does not depend on j. We obtain for any j, which, by Lemma 2, implies that the trees and are equal. ∎
Assume that the -term of equality (3.2) is nonzero. Then
If are the forest satisfying , , then Lemma 3 implies that there exists a cyclic permutation ρ on the roots of such that the ith tree of is equal to the th tree of , and thus . This implies that g can be reduced as a fraction for some permutations . Since the pair is already reduced, we obtain and , and thus all the terms in equality (3.2) are equal to zero except one. Therefore,
Therefore, Thompson’s group T has the Haagerup property.
Farley constructed a proper cocycle with values in a Hilbert space and showed in the proof of [4, Theorem 2.4] that if is described by a reduced pair of symmetric trees with n leaves, then . By Schoenberg’s theorem, this provides a family of positive definite functions
Note that if , then formula (3.3) implies for any . However, this equality is no longer true for certain elements of V. Consider
where is the full binary tree with four leaves. We have
We proved that T has the Haagerup property by using the net of maps , . One could hope to extend our proof using the same approximation of the identity for the larger group V. Unfortunately, the maps with are no longer vanishing at infinity if we consider them as functions on V. Indeed, consider the sequence of trees such that is the tree with two leaves and for . Note that is a tree with n leaves. Let be the tree equal to the composition of with a copy of attached to each leaf of . Define the permutation that is an involution and such that and for any , . Hence sends any odd leaf of the first copy of to the same leaf in the second copy of and lets invariant the others. We set and note that this fraction is reduced. If we consider , we observe that the term corresponding to in formula (3.2) is nonzero and is equal to . Since all the terms of are positive, we obtain for any , and thus does not tend to zero when g tends to infinity in V.
Funding source: H2020 European Research Council
Award Identifier / Grant number: 669240 QUEST
Funding source: Australian Research Council
Award Identifier / Grant number: DP140100732
Funding statement: A. Brothier was supported by European Research Council Advanced Grant 669240 QUEST and is now supported by a UNSW Sydney starting grant. V. F. R. Jones is supported by the grant numbered DP140100732, Symmetries of subfactors.
We thank the generous support of the New Zealand Mathematics Research Institute and the warm hospitality we received in Raglan, which made this work possible.
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