A NON-TRIVIAL EXAMPLE OF A FREE-BY-FREE GROUP WITH THE HAAGERUP PROPERTY FRANC ¸ OIS GAUTERO Abstract. The aim of this Note is to prove that the group of Formanek-Procesi acts properly isometrically on a finite dimensional CAT(0) cube complex. This gives a first example of a non-linear semidirect product between two non abelian free groups which satisfies the Haagerup property.

Introduction The Haagerup property is an analytical property introduced in [10], where it was proved to hold for free groups: Definition 1 ([10, 5]). A conditionally negative definite function on a discrete group 𝐺 is a function 𝑓 : 𝐺 → ℂ 𝑛 ∑ such that for any natural integer 𝑛, for any 𝜆1 , ⋅ ⋅ ⋅ , 𝜆𝑛 ∈ ℂ with 𝜆𝑖 = 0, for any 𝑖=1

𝑔1 , ⋅ ⋅ ⋅ , 𝑔𝑛 in 𝐺 one has ∑

𝜆𝑖 𝜆𝑗 𝑓 (𝑔𝑖−1 𝑔𝑗 ) ≤ 0.

𝑖,𝑗

The group 𝐺 satisfies the Haagerup property, or is an a-T-menable group, if and only if there exists a proper conditionnally negative definite function on 𝐺. Groups with the Haagerup property encompass the class of amenable groups, but form a much wider class. Free groups were in some sense the “simplest” non-amenable groups. Haagerup property has later been renewed by the work of Gromov, where it appeared under the term of a-T-menability. It is now most easily presented as a strong converse to the famous Kazhdan’s Property (T) in the sense that a group satisfies both the Haagerup and (T) properties if and only if it is a compact group (a finite group in the discrete case). We refer the reader to [5] for a detailed background and history of Haagerup property. What do we know about extensions of a-T-menable groups ? By [12] a semidirect product of an a-T-menable group with an amenable one is a-T-menable. For instance any semidirect product 𝔽𝑛 ⋊ ℤ, where 𝔽𝑛 denotes the rank 𝑛 free group, is a-T-menable. Also it has been proved recently that any wreath-product 𝔽𝑛 ≀ 𝔽𝑘 is a-T-menable (this is a particular case of the various theorems in [7] - see also the preliminary paper [6]). But such a result does not hold anymore when considering arbitrary Haagerup-by-Haagerup groups. The most famous counter-example is given by [3]: for any free subgroup 𝔽𝑘 of SL2 (ℤ) the semidirect product ℤ2 ⋊ 𝔽𝑘 satisfies a relative version of Kazhdans’s Property (T) and thus is not Haagerup (see also [8] for the relative property (T) of ℤ2 ⋊ SL(2, ℤ) and pass to a finite index free subgroup of SL2 (ℤ) - since the Haagerup property holds for Date: March 14, 2010. 2000 Mathematics Subject Classification. 20E22, 20F65, 20E05. Key words and phrases. Haagerup property, a-T-menability, free groups, semidirect products. 1

a group 𝐺 if and only if it holds for a finite index subgroup of 𝐺, this gives an example as announced). To what extent can this result be generalized to semidirect products 𝔽𝑛 ⋊ 𝔽𝑘 with both 𝑛 and 𝑘 greater or equal to 2 ? These semidirect products lie in some “philosophical” sense just “above” the groups ℤ2 ⋊𝔽𝑘 (substitute the amenable group ℤ2 by the free group 𝔽𝑛 , the simplest example of an a-T-menable but not amenable group) but also just above the groups 𝔽𝑛 ⋊ ℤ (substitute the free abelian group ℤ by the free non-abelian group 𝔽𝑘 ). The former analogy might lead to think that no group 𝔽𝑘 ⋊ 𝔽𝑛 (𝑛, 𝑘 ≥ 2) satisfies the Haagerup property whereas the latter one might lead to think that any such group is an a-T-menable group. The purpose of this short paper is to present a first example of a non-linear a-T-menable semidirect product 𝔽𝑛 ⋊ 𝔽𝑘 (𝑛, 𝑘 ≥ 2). More precisely: Definition 2. Let 𝑛 be any integer greater or equal to 2. The nth -group of Formanek - Procesi is the semidirect product 𝔽𝑛+1 ⋊𝜎 𝔽𝑛 where 𝔽𝑛 = ⟨𝑡1 , ⋅ ⋅ ⋅ , 𝑡𝑛 ⟩ and 𝔽𝑛+1 = ⟨𝑥1 , ⋅ ⋅ ⋅ , 𝑥𝑛+1 ⟩ are the rank 𝑛 and rank 𝑛 + 1 free groups and 𝜎 : 𝔽𝑛 ,→ Aut(𝔽𝑛+1 ) is the monomorphism defined as follows: For 𝑖 ∈ {1, ⋅ ⋅ ⋅ , 𝑛 + 1}, 𝜎(𝑡𝑖 )(𝑥𝑗 ) = 𝑥𝑗 if 𝑗 ∈ {1, ⋅ ⋅ ⋅ , 𝑛} and 𝜎(𝑡𝑖 )(𝑥𝑛+1 ) = 𝑥𝑛+1 𝑥𝑖 . As claimed by this definition, it is easily checked that 𝜎 is a monomorphism. These groups were introduced in [9] to prove that Aut(𝔽𝑛 ) is non linear for 𝑛 ≥ 3. Theorem 1. The nth -group of Formanek - Procesi acts properly isometrically on some (𝑛 + 2)dimensional CAT(0) cube complex and in particular satisfies the Haagerup property. Let us briefly recall that a cube complex is a metric polyhedral complex in which each cell is isomorphic to the Euclidean cube [0, 1]𝑛 and the gluing maps are isometries. A cube complex is called CAT(0) if the metric induced by the Euclidean metric on the cubes turns it into a CAT(0) metric space (see [2]). In order to get the above statement, we prove the existence of a “space with walls” structure as introduced by Haglund and Paulin [11]. A theorem of Chatterji-Niblo [4] or Nica [13] gives the announced action on a CAT(0) cube complex. Even if the proof is quite simple, we think however that our example is worth of interest: thanks to the profound structure theorem of [1] about subgroups of polynomially growing automorphisms (our example is a subgroup of linearly growing automorphisms), a more elaborated version of the construction presented here should hopefully lead to the proof of the following conjecture, well-known among experts in the field: Conjecture (folklore): Any semidirect product 𝔽𝑛 ⋊𝜎 𝔽𝑘 with 𝜎(𝔽𝑘 ) a free subgroup of polynomially growing outer automorphisms satisfies the Haagerup property. We would like in fact to propose a stronger conjecture: Conjecture: Any semidirect product 𝔽𝑛 ⋊𝜎 𝔽𝑘 with 𝜎(𝔽𝑘 ) a free subgroup of polynomially growing outer automorphisms acts on some 𝑁 -dimensional CAT(0) cube complex with 𝑟 ∑ 𝑁 ≤ 1+ max(rank(𝐻𝑖+1 ) − rank(𝐻𝑖 ), 1) where 𝑟 is the number of strata in the invari𝑖=1

ant filtration of 𝔽𝑛 for 𝜎(𝔽𝑘 ) given by the Bestvina-Feighn-Handel theorem of [1], and the 𝐻𝑖 ’s are the strata of this filtration: ∅ ⊈ 𝐻1 ⊈ 𝐻2 ⊈ ⋅ ⋅ ⋅ ⊈ 𝐻𝑟 = 𝔽𝑛 . The proposed upper-bound in the Conjecture above is only a rough guess, see the brief discussion at the end of the paper. 2

1. Preliminaries 1.1. Notations. We will prove Theorem 1 with 𝑛 = 2. The reader will easily generalize the construction to any integer 𝑛 ≥ 2. With the notations of Theorem 1, the group 𝐺 := 𝔽3 ⋊𝜎 𝔽2 admits −1 ⟨𝑥𝑖 , 𝑡𝑗 ; 𝑡−1 𝑗 𝑥𝑗 𝑡𝑗 = 𝑥𝑗 , 𝑡𝑗 𝑥3 𝑡𝑗 = 𝑥3 𝑥𝑗 , 𝑖 = 1, 2, 3 and 𝑗 = 1, 2⟩

as a presentation. We denote by 𝑆 the generating set {𝑥1 , 𝑥2 , 𝑥3 , 𝑡1 , 𝑡2 } of 𝐺. In the structure of semidirect product 𝔽3 ⋊𝜎 𝔽2 we will term horizontal subgroup the normal subgroup 𝔽3 = ⟨𝑥1 , 𝑥2 , 𝑥3 ⟩ and vertical subgroup the subgroup 𝔽2 = ⟨𝑡1 , 𝑡2 ⟩. Any element is uniquely written as a concatenation 𝑡𝑤 where 𝑡 is a vertical element, i.e. an element in the vertical subgroup, and 𝑤 is a horizontal element, i.e. an element in the horizontal subgroup. We denote by 𝒜 the alphabet over 𝑆 ∪ 𝑆 −1 and by 𝜋 the map which, to a given word in 𝒜, assigns the unique element of 𝐺 that it defines. A reduced word is a word without any cancellation 𝑥𝑥−1 or 𝑥−1 𝑥. Words defining vertical (resp. horizontal) elements are vertical (resp. horizontal) words. Definition 1.1. A standard form for an element 𝑔 in 𝐺 is a word in the alphabet 𝒜 of the form (𝑣1 ⋅ ⋅ ⋅ 𝑣𝑘 ℎ1 ⋅ ⋅ ⋅ ℎ𝑟 )±1 whose image under 𝜋 is 𝑔, where each 𝑣𝑖 (resp. each ℎ𝑖 ) is a reduced vertical (resp. horizontal) word and no cancellation occurs between two consecutive vertical (resp. horizontal) words 𝑣𝑖 𝑣𝑖+1 (resp. ℎ𝑗 ℎ𝑗+1 ), 𝑖 = 1, ⋅ ⋅ ⋅ , 𝑘 − 1 and 𝑗 = 1, ⋅ ⋅ ⋅ , 𝑟 − 1. We denote by Γ the Cayley graph of 𝐺 with respect to 𝑆. Since the vertices of Γ are in bijection with the elements of 𝐺, we do not distinguish between a vertex of Γ and the element of 𝐺 associated to this vertex. An edge of Γ is denoted by the pair “(initial vertex of the edge, terminal vertex of the edge)”. The edges are labeled with the elements in 𝑆 ∪ 𝑆 −1 . For instance the edge (𝑔, 𝑔𝑥𝑖 ) has label 𝑥𝑖 , whereas the edge (𝑔𝑥𝑖 , 𝑔) has label 𝑥−1 𝑖 . If 𝑥 is the label of an edge we will term this edge 𝑥-edge. When considering Γ as a cellular complex, there is exactly one 1-cell associated to the two edges (𝑔, 𝑔𝑥𝑖 ) and (𝑔𝑥𝑖 , 𝑔), and each orientation of this 1-cell corresponds to one of these edges. Lemma 1.2. With the notations above, the group 𝐺 admits 𝑆min := {𝑥3 , 𝑡1 , 𝑡2 } ⊂ 𝑆 as a generating set. −1 Proof. For 𝑖 ∈ {1, 2} we have 𝑥𝑖 = 𝑥−1 3 𝑡𝑖 𝑥3 𝑡𝑖 hence the lemma.

□

As a straightforward consequence: Corollary 1.3. Let 𝜒𝑖 be the set of 1-cells of Γ associated to edges with label 𝑥±1 and let Γ𝑐 be the 𝑖 closure of the complement of 𝜒1 ∪ 𝜒2 in Γ. Then Γ𝑐 is (𝐺-equivariantly homeomorphic to) the Cayley graph of 𝐺 with respect to the generating set 𝑆min defined in Lemma 1.2. 1.2. Space with walls structure. In order to check the Haagerup property, in [11] were introduced the spaces with walls. A space with walls is a pair (𝑋, 𝒲) where 𝑋 is a set and 𝒲 is a family of partitions of 𝑋 into two classes, called walls, such that for any two distinct points 𝑥, 𝑦 in 𝑋 the number of walls 𝜔(𝑥, 𝑦) is finite. This is the wall distance between 𝑥 and 𝑦. We say that a discrete groups acts properly on a space with walls (𝑋, 𝒲) if it leaves invariant 𝒲 and for some (and hence any) 𝑥 ∈ 𝑋 the function 𝑔 7→ 𝜔(𝑥, 𝑔𝑥) is proper on 𝐺. 3

Theorem 1.4 ([11]). A discrete group 𝐺 which acts properly on a space with walls satisfies the Haagerup property. In order to get Theorem 1 we will need the following (stronger) result (we refer to [13] for a similar statement): Theorem 1.5 ([4]). Let 𝐺 be a discrete group which acts properly on a space with walls (𝑋, 𝒲). Say that two walls (𝑢, 𝑢𝑐 ) ∈ 𝒲 and (𝑣, 𝑣 𝑐 ) ∈ 𝒲 cross if all four intersections 𝑢 ∩ 𝑣, 𝑢 ∩ 𝑣 𝑐 , 𝑢𝑐 ∩ 𝑣 and 𝑢𝑐 ∩ 𝑣 𝑐 are non-empty. Let 𝐼(𝒲) be the supremum of the cardinalities of finite collections of walls which pairwise cross and assume that 𝐼(𝒲) is finite. Then 𝐺 acts properly isometrically on some 𝐼(𝒲)-dimensional CAT(0) cube complex. In particular it satisfies the Haagerup property. 2. Horizontal and vertical walls 2.1. Definition and stabilizers. Definition 2.1. The horizontal block 𝒳3 is the set of all the elements in 𝐺 which admit 𝑡𝑥3 𝑤, with 𝑡 a vertical word and 𝑤 a horizontal word, as a standard form. A horizontal wall is a left-translate 𝑔(𝒳3 , 𝒳3𝑐 ), 𝑔 ∈ 𝐺. The vertical 𝑗-block 𝒱𝑗 is the set of all the elements in 𝐺 which admit 𝑡𝑗 𝑡𝑤, with 𝑡 a vertical word and 𝑤 a horizontal word, as a standard form. A vertical 𝑗-wall is a left-translate 𝑔(𝒱𝑗 , 𝒱𝑗𝑐 ), 𝑔 ∈ 𝐺. By definition of a standard form, in the definition of 𝒳3 (resp. of 𝒱𝑗 ), 𝑤 (resp. 𝑡) does −1 not begin by 𝑥−1 3 (resp. by 𝑡𝑗 ). Lemma 2.2. The collection of either all the horizontal or all the vertical walls is 𝐺-invariant for the left-action of 𝐺 on itself. Moreover: (1) The left 𝐺-stabilizer of any horizontal wall is a conjugate of the vertical subgroup. (2) The horizontal subgroup is both the left and right 𝐺-stabilizer of any vertical wall. Proof. By definition the collection of either all the horizontal or all the vertical walls consists of all the left 𝐺-translates of the horizontal or vertical walls (𝒳3 , 𝒳3𝑐 ) or (𝒱𝑗 , 𝒱𝑗𝑐 ) so that it is invariant under the left 𝐺-action. Let 𝑔 ∈ 𝒳3 . Then 𝑔 = 𝑡𝑥3 𝑤 for some 𝑡 in the vertical subgroup and 𝑤 in the horizontal one. If 𝑡′ is another element in the vertical subgroup, 𝑡′ 𝑔 = 𝑡′ 𝑡𝑥3 𝑤 ∈ 𝒳𝑖 . Thus the vertical subgroup is in the 𝐺-stabilizer of 𝒳3 . By the relation 𝑢𝑡 = 𝑡𝜎(𝑡)(𝑢) for 𝑢 in the horizontal subgroup (recall that 𝜎 : 𝔽2 ,→ Aut(𝔽3 ) is the monomorphism such that 𝐺 = 𝔽3 ⋊𝜎 𝔽2 ) we get 𝑢𝑔 = 𝑡𝜎(𝑡)(𝑢)𝑥3 𝑤 so that ⟨𝑢⟩ does not stabilize 𝒳3 . Since any element of 𝐺 is the concatenation of a vertical element with a horizontal one, these observations imply that the 𝐺-stabilizer of 𝒳3 is the vertical subgroup. Since the horizontal walls are left 𝐺-translates of the wall (𝒳3 , 𝒳3𝑐 ), the 𝐺-stabilizer of a horizontal wall is a conjugate of the vertical subgroup. The proof for the stabilizers of the vertical walls are similar and easier: just observe that since the horizontal subgroup is normal in 𝐺, it is useless to take its conjugates. □ 4

2.2. Finiteness of horizontal and vertical walls between any two elements. Proposition 2.3. There are a finite number of vertical walls between any two elements. Proof. The vertical walls are the usual walls used to prove that the free group 𝔽2 satisfies the Haagerup property. Thus there are a finite number (in fact one) of vertical walls between 𝑔 ∈ 𝐺 and 𝑔𝑡𝑖 with 𝑡𝑖 a vertical generator. By Lemma 2.2 each vertical wall is stabilized by the right-action of the horizontal subgroup. Thus no vertical wall separates 𝑔 from 𝑔𝑥𝑖 , 𝑔 ∈ 𝐺 and 𝑥𝑖 a horizontal generator. The proposition follows. □ Proposition 2.4. There are a finite number of horizontal walls between any two elements in 𝐺. This proposition is a little bit more difficult than the previous one and we need a preliminary lemma: Lemma 2.5. Horizontal walls are invariant under the right-action of the vertical subgroup. Proof. We begin by the Claim 1. Let 𝑋3 be the intersection of 𝒳3 with the horizontal subgroup 𝔽3 . Then 𝑋3 is invariant under the right-action of the vertical subgroup 𝔽2 . Proof. If 𝑔 ∈ 𝑋3 then 𝑔 = 𝑥3 𝑤 with 𝑤 in 𝔽3 and the reduced form of 𝑤 does not begin with 𝑥−1 3 . If 𝑡 is any element in the vertical subgroup then 𝑔.𝑡 = 𝑥3 𝑤.𝑡 = 𝜎(𝑡)(𝑥3 )𝜎(𝑡)(𝑤) = 𝑥3 𝑤1 where 𝑤1 is a horizontal word such that 𝑥3 𝑤1 is reduced. Indeed, for any 𝑗 = 1, 2 one has 𝑥3 .𝑡±1 = 𝑥3 𝑥±1 so that no cancellation occurs which might suppress the letter 𝑥3 𝑗 𝑗 in the concatenation 𝜎(𝑡)(𝑥3 )𝜎(𝑡)(𝑤) = 𝜎(𝑡)(𝑥3 𝑤) (recall that 𝑥3 𝑤 was reduced). This proves the claim. □ ∪ ∪ It follows from the claim that (𝒳3 , 𝒳3𝑐 ) = 𝑡(𝑋3 , 𝑋3𝑐 ) = (𝑋3 , 𝑋3𝑐 )𝑡. Therefore 𝑡∈𝔽2

𝑡∈𝔽2

𝑢(𝒳3 , 𝒳3𝑐 )𝑡 = 𝑢(𝒳3 , 𝒳3𝑐 ) for any 𝑡 in the vertical subgroup and for any 𝑢 in the horizontal one. Lemma 2.5 is proved. □ Proof of Proposition 2.4. By Lemma 2.5, there is no horizontal wall between any two elements 𝑔 and 𝑔𝑡𝑗 , 𝑗 = 1, 2. Moreover there is exactly one horizontal wall between any two elements 𝑔 and 𝑔𝑥3 . This is indeed true if one considers the intersection of the set of horizontal walls with the horizontal subgroup: the set of walls so obtained is a subset of the classical set of walls for the free group. The right-invariance under the action of the vertical subgroup given by Lemma 2.5 gives the initial claim. We so proved that there is a finite number of walls between any two elements 𝑔 and 𝑔𝑥3 and between any two elements 𝑔 and 𝑔𝑡𝑗 . By Lemma 1.2 we get the finiteness of the number of walls between any two elements in 𝐺. □ Before beginning the next section, we recall that Γ𝑐 denotes the Cayley graph of 𝐺 with respect to 𝑆min = {𝑥3 , 𝑡1 , 𝑡2 } (see Lemma 1.2). 5

3. Vertizontal walls 3.1. Definition and stabilizers. Lemma 3.1.∪ 𝑘 −𝑘+1 ±1 Let 𝐸𝑖 := ) (𝑖 = 1, 2 mod 2), let 𝑀𝑖 be the set of the midpoints of (𝑥𝑘𝑖 𝑡−𝑘 𝑖 , 𝑥𝑖 𝑡𝑖 𝑘∈ℤ

the 1-cells in 𝐸𝑖 and let ℳ𝑖 := ⟨𝑥𝑖+1 , 𝑡𝑖+1 , 𝑥3 𝑥𝑖 𝑥−1 3 𝑡𝑖 ⟩𝑀𝑖 . Let 𝒯𝑖 denote the set of all the elements in 𝐺 which are connected to the identity vertex 𝑒 by an edge-path in Γ𝑐 ∖ ℳ𝑖 . Then the 𝑖-block 𝒯𝑖 is neither empty nor equal to 𝐺. More precisely its complement 𝒯𝑖𝑐 is the non-empty set of all the elements in 𝐺 which are connected to 𝑡𝑖 by an edge-path in Γ𝑐 ∖ ℳ𝑖 . See Figure 1. Beware that this figure might be slightly misleading when considering the edge-paths 𝑥3 𝑥𝑖 𝑥−1 3 : they are indeed preserved under the right-action of the vertical subgroup but this is a consequence of the fact that a cancellation occurs between −1 −1 𝜎(𝑡𝑖 )(𝑥𝑖 ) = 𝑥𝑖 and 𝜎(𝑡𝑖 )(𝑥−1 3 ) = 𝑥𝑖 𝑥3 . Since we did not draw the images of the edges in these edge-paths, this cancellation does not appear in the figure.

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ti i = 1,2 Figure 1. The ⟨𝑥3 𝑥𝑖 𝑥−1 3 𝑡𝑖 ⟩-translation of 𝐸𝑖 6

ti

Proof. We fix 𝑖 in {1, 2} modulo 2. By definition 𝒯𝑖 contains the identity element 𝑒. Let us prove that 𝑡𝑖 ∈ / 𝒯𝑖 . Since there are no 𝑥±1 𝑖 -edge in Γ𝑐 the set 𝐸𝑖 disconnects the subset of Γ𝑐 associated to ⟨𝑥𝑖 , 𝑡𝑖 ⟩. Moreover 𝑒 and 𝑡𝑖 belong to two distinct connected components in this grid. Since there are no 𝑥±1 𝑖+1 -edge it is impossible to connect 𝑒 to 𝑡𝑖 in Γ𝑐 without leaving 𝒯𝑖 and going through some 𝑥±1 3 -edge. Let us denote by 𝑝 the edge-path one is looking for, from 𝑒 to 𝑡𝑖 . Since one uses the left ⟨𝑥𝑖+1 , 𝑡𝑖+1 ⟩-translates of 𝐸𝑖 to disconnect Γ𝑐 , we can assume that 𝑝 goes only through ±1 ±1 𝑥±1 3 - and 𝑡𝑖 -edges. The first horizontal edge in 𝑝 is some 𝑥3 -edge the initial vertex of −𝑚 which has the form 𝑡𝑖 with 𝑚 a positive integer. ±1 Assume that the first horizontal edge in 𝑝 is a 𝑥−1 = 3 -edge. Recall the relations 𝑥3 𝑡𝑖 ±1 ±1 𝑡𝑖 𝑥3 𝑥𝑖 . Then, from Lemma 2.5 and by the tautology that horizontal walls separate Γ𝑐 in two components, 𝑝 has to eventually go through some 𝑥−1 3 -edge the terminal vertex of which is the initial vertex of some edge in 𝐸𝑖 . But 𝑡𝑖 is connected in Γ𝑐 ∖ ℳ𝑖 to the 𝑘 terminal vertex of this edge by the edge-path 𝑥−1 3 𝑡𝑖 𝑥3 . We are so back to the initial problem. Assume now that the first horizontal edge in 𝑝 is a 𝑥3 -edge. Then we can assume that 𝑚 the last horizontal edge in 𝑝 is a 𝑥−1 3 -edge ending with some element 𝑡𝑖 with 𝑚 a positive ±1 ±1 ±1 integer. Since 𝑥3 𝑡𝑖 = 𝑡𝑖 𝑥3 𝑥𝑖 , 𝑝 has to go through the orbit, under the right action of the vertical subgroup, of the 𝑥𝑖 -edge between 𝑥3 𝑥−1 ∈ 𝐺 and 𝑥3 ∈ 𝐺. But one is 𝑖 looking for 𝑝 ⊂ Γ𝑐 ∖ ℳ𝑖 . We so get a contradiction with the fact that ℳ𝑖 contains the left ⟨𝑥3 𝑥𝑖 𝑥−1 3 𝑡𝑖 ⟩-translates of the midpoints of the 1-cells in 𝐸𝑖 . We so proved that 𝑡𝑖 ∈ / 𝒯𝑖 . It is easy to see that the initial vertices of the edges in −1 ⟨𝑥3 𝑥𝑖 𝑥3 𝑡𝑖 ⟩𝐸𝑖 belong to 𝒯𝑖 whereas their terminal vertices belong to 𝒯𝑖𝑐 . Indeed edge−1 𝑘 paths reading (𝑥−1 3 𝑡𝑖 𝑥3 ) , 𝑘 ∈ ℤ, connect the initial (resp. terminal) vertices of 𝐸𝑖 to 𝑒 𝑘 (resp. to 𝑡𝑖 ) in Γ𝑐 ∖ℳ𝑖 whereas edge-paths reading (𝑥3 𝑡𝑖 𝑥−1 3 ) connect 𝑒 (resp. 𝑡𝑖 ) to initial −1 (resp. terminal) vertices in the various left ⟨𝑥3 𝑥𝑖 𝑥3 𝑡𝑖 ⟩-translates of 𝐸𝑖 . We conclude the proof of the lemma by noticing that an element that cannot be connected neither to 𝑒 nor 𝑡𝑖 in Γ𝑐 ∖ ℳ𝑖 cannot be connected to any element in any ⟨𝑥𝑖+1 , 𝑡𝑖+1 , 𝑥3 ⟩-translate of ⟨𝑥𝑖 , 𝑡𝑖 ⟩ (𝑖 = 1, 2 mod 2). Since we so get the whole group 𝐺, we get a contradiction and □ the description of 𝒯𝑖𝑐 as announced by the lemma. By Lemma 3.1 the following definition makes sense: Definition 3.2. A vertizontal 𝑖-wall (𝑖 = 1, 2) is any left-translate 𝑔(𝒯𝑖 , 𝒯𝑖𝑐 ) where 𝑔 ∈ 𝐺 and 𝒯𝑖 is the 𝑖-block defined in Lemma 3.1. Lemma 3.3. The collection of all the vertizontal 𝑖-walls (𝑖 = 1, 2) is 𝐺-invariant for the left-action of 𝐺 onto itself. The left 𝐺-stabilizer of any vertizontal 𝑖-wall (𝑖 = 1, 2 mod 2) is a −1 conjugate of the subgroup 𝐻𝑖 = ⟨𝑥𝑖+1 , 𝑡𝑖+1 , 𝑥3 𝑥𝑖 𝑥−1 3 𝑡𝑖 , 𝑥𝑖 𝑡𝑖 ⟩. Proof. The left 𝐺-invariance is obvious, as in the proof of Lemma 2.2. Let us check the assertion about the left 𝐺-stabilizers. A vertizontal wall is a left 𝐺-translate of (𝒯𝑖 , 𝒯𝑖𝑐 ), where 𝒯𝑖 is the 𝑖-block defined in Lemma 3.1. Thus its left 𝐺-stabilizer is conjugate in 𝐺 to the left 𝐺-stabilizer of (𝒯𝑖 , 𝒯𝑖𝑐 ), denoted by Stab𝐺 (𝒯𝑖 , 𝒯𝑖𝑐 ). Since 𝒯𝑖 is separated from 𝒯𝑖𝑐 by the left ⟨𝑥𝑖+1 , 𝑡𝑖+1 , 𝑥3 𝑥𝑖 𝑥−1 3 𝑡𝑖 ⟩-translate of 𝐸𝑖 (see 𝑐 Lemma 3.1) we have ⟨𝑥𝑖+1 , 𝑡𝑖+1 , 𝑥3 𝑥𝑖 𝑥−1 𝑡 , Stab (𝐸 )⟩ = Stab (𝒯 𝐺 𝑖 𝐺 𝑖 , 𝒯𝑖 ). Since Stab𝐺 (𝐸𝑖 ) = 3 𝑖 𝑐 ⟨𝑥𝑖 𝑡−1 □ 𝑖 ⟩ we get 𝐻𝑖 = Stab𝐺 (𝒯𝑖 , 𝒯𝑖 ). 7

3.2. Finiteness of the vertizontal walls between any two elements. Proposition 3.4. There are a finite number of vertizontal walls between any two elements in 𝐺. Proof. We consider the set of vertizontal 1-walls (the proof is the same for the vertizontal 2-walls). We work with the generating set 𝑆min = {𝑥3 , 𝑡1 , 𝑡2 } given by Lemma 1.2. Since ±1 −1 any 𝑥±1 3 - and any 𝑡2 -edge lies in Γ𝑐 ∖ ⟨𝑥𝑖+1 , 𝑡𝑖+1 , 𝑥3 𝑥𝑖 𝑥3 𝑡𝑖 ⟩𝐸𝑖 (see the notation of Lemma 3.1) no vertizontal 1-wall is intersected when passing from 𝑔 to 𝑔𝑥3 nor from 𝑔 to 𝑔𝑡2 . Thus one only has to check which vertizontal 1-walls are intersected when passing from 𝑒 to 𝑡1 . There is of course the wall (𝒯1 , 𝒯1𝑐 ). Assume there is another wall 𝑔(𝒯1 , 𝒯1𝑐 ). Then, by definition, this wall corresponds to the partition of Γ𝑐 in two components given ±1 ±1 by 𝑔⟨𝑥𝑖+1 , 𝑡𝑖+1 , 𝑥3 𝑥𝑖 𝑥−1 ∈ 𝑔⟨𝑥𝑖+1 , 𝑡𝑖+1 , 𝑥3 𝑥𝑖 𝑥−1 ∈ 3 𝑡𝑖 ⟩𝐸𝑖 . Thus (𝑒, 𝑡1 ) 3 𝑡𝑖 ⟩𝐸𝑖 . But (𝑒, 𝑡1 ) −1 −1 𝑐 𝐸𝑖 . This implies 𝑔⟨𝑥𝑖+1 , 𝑡𝑖+1 , 𝑥3 𝑥𝑖 𝑥3 𝑡𝑖 ⟩𝐸𝑖 = ⟨𝑥𝑖+1 , 𝑡𝑖+1 , 𝑥3 𝑥𝑖 𝑥3 𝑡𝑖 ⟩𝐸𝑖 so that 𝑔(𝒯1 , 𝒯1 ) = □ (𝒯1 , 𝒯1𝑐 ) and we are done. 4. A proper action We prove the following Proposition 4.1. Let 𝒲 be the set of all the horizontal, vertical and vertizontal walls. The left-action of 𝐺 on itself defines a proper action on the space with walls structure (𝐺, 𝒲). Proof. Each element 𝑔 ∈ 𝐺 has a standard form 𝑤𝑡 with 𝑤 a reduced horizontal word and 𝑡 a reduced vertical word. Since the vertical walls are the classical walls in the free group 𝔽2 , the number of vertical walls intersected goes to infinity with the number of letters in 𝑡. Thus we can assume that 𝑔 admits 𝑤 as a standard form. From Lemma 2.5 the number of horizontal walls intersected goes to infinity with the number of 𝑥±1 3 -letters in 𝑤. Thus we can assume that 𝑔 admits a standard form containing only 𝑥±1 𝑖 -letters, 𝑖 = 1, 2. Observe that in order to go from 𝑒 to 𝑥𝑖 in Γ𝑐 one has to intersect two vertizontal 𝑐 𝑖-walls: the 𝑖-wall (𝒯𝑖 , 𝒯𝑖𝑐 ) and the 𝑖-wall 𝑥−1 3 (𝒯𝑖 , 𝒯𝑖 ). These are indeed the two 𝑖-walls −1 intersected by 𝑥−1 3 𝑡𝑖 𝑥3 𝑡𝑖 . Thus the number of walls intersected goes to infinity with the ±1 𝑘 number of times the letters 𝑥±1 1 and 𝑥2 alternate in 𝑤. Therefore we can assume 𝑤 = 𝑥1 −𝑘 𝑘 𝑘 with 𝑘 ∈ ℕ. The edge-path 𝑥−1 3 𝑡1 𝑥3 𝑡1 goes from 𝑒 to 𝑥1 in Γ𝑐 . It intersects 2𝑘 vertizontal 2 𝑘−1 1-walls which are the left-translates by 𝑥1 , 𝑥1 , ⋅ ⋅ ⋅ , 𝑥 of the vertizontal 1-walls (𝒯1 , 𝒯1𝑐 ) 𝑐 and 𝑥−1 □ 3 (𝒯1 , 𝒯1 ) given above. We so get the lemma. 5. Conclusion By Propositions 4.1 and 3.4, 𝐺 acts properly on a space with walls structure (𝐺, 𝒲) where 𝒲 is the set of all the horizontal, vertical and vertizontal walls. In particular by [11] it satisfies the Haagerup property. By [4] it acts on a 𝐼(𝒲)-dimensionnal cube complex where 𝐼(𝒲) is the supremum of the cardinalities of finite collections of walls which pairwise cross (see Theorem 1.5). Obviously here there exists such a collection which is composed of four walls: the horizontal wall (𝒳3 , 𝒳3𝑐 ), the vertical wall (𝒱1 , 𝒱1𝑐 ), the vertizontal 1-wall (𝒯1 , 𝒯1𝑐 ) and the vertizontal 2-wall (𝒯2 , 𝒯2𝑐 ). As soon as there is a vertical wall in such a collection - recall that these are the classical walls of the free group - no other vertical wall can be added. The same is true for the horizontal walls since their intersections with the horizontal subgroup are the classical walls of the free group and by Lemma 2.5 they are invariant under the right-action of the vertical subgroup. 8

It remains to check that this also holds when adding a vertizontal 𝑖-wall to a collection already containing such a vertizontal 𝑖-wall. But it suffices to look at what happens for the intersection of these walls with the ⟨𝑥𝑖 , 𝑡𝑖 ⟩-subgroups (𝑖 = 1, 2): one is “higher” than the other so that the connected component at the top of one does not intersect the connected component at the bottom of the other. The proof of Theorem 1 is complete, at least in the case where 𝑛 = 2. As we noticed the generalization to any integer 𝑛 ≥ 3 is straightforward: vertical walls are the same, horizontal walls correspond to the orbits of 𝑥𝑛+1 under the vertical subgroup, vertizontal walls are associated to 𝑡1 , ⋅ ⋅ ⋅ , 𝑡𝑛 . □ Remark 5.1. We briefly come back to the Conjecture proposed at the end of the Introduction. By adapting our construction we get that the group 𝔽𝑛+1 ⋊𝜎 𝔽2 (𝑛 ≥ 3), where 𝜎(𝑡𝑖 ) fixes any 𝑥𝑗 for 𝑗 = 1, ⋅ ⋅ ⋅ , 𝑛 and 𝜎(𝑡𝑖 )(𝑥𝑛+1 ) = 𝑥𝑛+1 𝑥𝑖 , 𝑖 = 1, 2, acts on a 4-dimensional CAT(0) cube complex: the walls are the vertical and vertizontal walls as defined above, and horizontal walls associated not only to 𝑥𝑛+1 but also to 𝑥3 , ⋅ ⋅ ⋅ , 𝑥𝑛 . There are more types of horizontal walls but less types of vertizontal walls than in the nth -group of Formanek - Procesi. Since two distinct horizontal walls cannot be in a collection of pairwise crossing walls (contrary to what happens with vertizontal walls) this explains the smaller dimension of the complex in this case. Thus, perhaps more important than the number of strata or the difference of the ranks in two consecutive strata of the filtration is the way the images of the higher edges cover the lower strata. Here the rose with 𝑛 + 1 petals is a Bestvina-Feighn-Handel representative. The filtration of the graph is given by ∅ ⊈ {𝑥1 , ⋅ ⋅ ⋅ , 𝑥𝑛 } ⊈ {𝑥1 , ⋅ ⋅ ⋅ , 𝑥𝑛 } ∪ {𝑥𝑛+1 }. The image of the highest edge 𝑥𝑛+1 cover {𝑥1 , 𝑥2 } but not the whole lower stratum {𝑥1 , ⋅ ⋅ ⋅ , 𝑥𝑛 } as it is the case when considering the nth -group of Formanek - Procesi. Acknowledgements: The author would like to thank Martin Lustig (Marseille) for giving to him, while working on other topics, the example of free-by-free group we deal with in this paper. It is also a pleasure to thank P.A. Cherix (Gen`eve) who introduced the author to the Haagerup property almost seven years ago, and together with G.N. Arzhantseva (Neuchatel) who evoked the question of the a-T-menability of free-by-free, and surfaceby-free groups. The author is also indebted to G.N. Arzhantseva for telling him about the Conjecture cited in the Introduction. G.N. Arzhantseva, M. Lustig and Y. Stalder also deserve many thanks for their remarks about preliminary versions of the paper. Finally, the referee of a first version of this paper gave invaluable comments to make it better: many thanks are due to him. References [1] Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(𝐹𝑛 ). II. A Kolchin type theorem. Annals of Mathematics, 161(1):1–59, 2005. [2] Martin R. Bridson and Andr´e Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. [3] M. Burger. Kazhdan constants for SL(3, Z). Journal f¨ ur die Reine und Angewandte Mathematik, 413:36–67, 1991. [4] Indira Chatterji and Graham Niblo. From wall spaces to CAT(0) cube complexes. International Journal of Algebra and Computation, 15(5-6):875–885, 2005. [5] Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette. Groups with the Haagerup property, volume 197 of Progress in Mathematics. Birkh¨auser Verlag, Basel, 2001. [6] Yves de Cornulier, Yves Stalder, and Alain Valette. Proper actions of lamplighter groups associated with free groups. Comptes Rendus de l’Acad´emie des Sciences, 346(3-4):173–176, 2008. 9

[7] Yves de Cornulier, Yves Stalder, and Alain Valette. Proper action of wreath products and generalizations. 2009. arXiv:0905.3960v1[math.GR]. [8] Pierre de la Harpe and Alain Valette. La propri´et´e (𝑇 ) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger). Ast´erisque, (175):158, 1989. [9] Edward Formanek and Claudio Procesi. The automorphism group of a free group is not linear. Journal of Algebra, 149(2):494–499, 1992. [10] Uffe Haagerup. An example of a nonnuclear 𝐶 ∗ -algebra, which has the metric approximation property. Inventiones Mathematicae, 50(3):279–293, 1978/79. [11] Fr´ed´eric Haglund and Fr´ed´eric Paulin. Simplicit´e de groupes d’automorphismes d’espaces `a courbure n´egative. In The Epstein birthday schrift, volume 1 of Geom. Topol. Monogr., pages 181–248 (electronic). Geom. Topol. Publ., Coventry, 1998. [12] Paul Jolissaint. Borel cocycles, approximation properties and relative property T. Ergodic Theory and Dynamical Systems, 20(2):483–499, 2000. [13] Bogdan Nica. Cubulating spaces with walls. Algebraic & Geometric Topology, 4:297–309 (electronic), 2004. ´ de Nice Sophia Antipolis, Laboratoire de Mathe ´matiques Franc ¸ ois Gautero, Universite ´ J.A. Dieudonne (UMR CNRS 6621), Parc Valrose, 06108 Nice Cedex 2, France E-mail address: [email protected]

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