FREE-BY-FREE GROUPS OVER POLYNOMIALLY GROWING AUTOMORPHISMS ARE A-T-MENABLE FRANC ¸ OIS GAUTERO Abstract. We prove that any group G = Fn oσ Fk , where Fn and Fk are respectively rank n and rank k free groups, and σ : Fk ,→ Aut(Fn ) is a monomorphism such that σ(Fk ) is a subgroup consisting entirely of polynomially growing automorphisms of Fn , acts properly isometrically on a finite dimensional CAT(0) cube complex. This is achieved by exhibiting a proper space-with-walls structure in the sense of Haglund-Paulin. In particular any such group G is a-T-menable in the sense of Gromov (equivalently satisfies the Haagerup property).

Introduction The Haagerup property is an analytical property on groups introduced in [11], about the existence of a proper conditionally negative definite function: Definition 1. A conditionally negative definite function on a topological group G is a function f : G → R such that for any positive integer n, for any λ1 , · · · , λn ∈ R with n X λi = 0, for any g1 , · · · , gn in G one has i=1

X

λi λj f (gi−1 gj ) ≤ 0.

i,j

A function f : G → R on a topological group G is proper if lim f (gi ) = ∞. gi →∞

In a topological group G a sequence of elements (gi )i∈N ⊂ G tends to infinity with i if and only if for any compact K ⊂ G there is N ≥ 0 such that for any i ≥ N , gi does not belong to K. In this paper we will work with discrete groups, that is groups with the discrete topology, equipped with a word-length: let us recall that, if G is a group with generating set S, the word-length of γ ∈ G with respect to S, denoted by |γ|S , is the minimum of the lengths of the words in S ∪ S −1 which define the element γ. Saying that a sequence (gi )i∈N ⊂ G tends to infinity with i then amounts to saying that the word-lengths of the gi ’s tend to infinity with i. In [11] Haagerup property was proved to hold for free groups. It has later been renewed by the work of Gromov, where it appeared under the term of a-T-menability. The origin of this terminology is that on the one hand any amenable group satisfies this property and on the other hand it is a weak converse to Kazhdan’s property T: any group which satisfies both properties is a compact group (a finite group in the discrete case). Definition 2 ([11, 7, 1]). The group G satisfies the Haagerup property, or is an a-Tmenable group, if and only if there exists a proper conditionally negative definite function on G. Date: October 4, 2012. 2000 Mathematics Subject Classification. 20E22, 20F65, 20E05. Key words and phrases. Haagerup property, a-T-menability, free groups, semidirect products, improved relative train-track maps. 1

We refer the reader to [7] for a detailed background and history of this property. Let us notice that very few is known about the preservation of a-T-menability under extensions: if any semi-direct product (a-T-menable) o (amenable) is a-T-menable [13], this is not the case for any semi-direct product (a-T-menable) o (a-T-menable). For instance Z2 o SL(2, Z), which has the form (amenable) o (a-T-menable) since SL(2, Z) admits a free subgroup of finite index (a-T-menabilty passes from a finite index subgroup of a group to the group itself), has relative property (T) and so is not a-T-menable (any conditionally negative definite function is bounded on Z2 , see [8] - in fact for any free subgroup Fk of SL(2, Z), Z2 o Fk is not a-T-menable - see [5]). In particular it is not known whether any free-by-free group is a-T-menable. In [9] we exhibit a first family of groups of the form (free non abelian)-by-(free non abelian), termed Formanek-Procesi groups, which are a-T-menable. More precisely a Formanek-Procesi group is a semi-direct product Fn oσ Fn−1 with Fn = hx1 , · · · , xn i and Fn−1 = ht1 , · · · , tn−1 i free non-abelian groups respectively of ranks n and n − 1 and where σ(Fn−1 ) < Aut(Fn ) (Aut(Fn ) is the group of automorphisms of Fn ) is defined by: σ(tj )(xi ) = xi for any i = 1, · · · , n − 1 and j = 1, · · · , n − 1 whereas σ(tj )(xn ) = xn xj for j = 1, · · · , n − 1. A particularity of σ(Fn−1 ) is to have linear growth: Definition 3. An automorphism α of a group G with finite generating S has polynomial growth if and only if there is a polynomial function P such that, for any γ in G, for any m ∈ Z, |αm (γ)|S ≤ P (m)|γ|S . The growth is linear when the polynomial function has degree one. The purpose of this paper is to prove that the result of [9] is a particular case of a much more general phenomenon: Theorem 1. Let k, n be two positive integers and let Fn , Fk respectively denote the rank n and rank k free groups. Let Aut(Fn ) denote the group of automorphisms of Fn . If σ : Fk ,→ Aut(Fn ) is a monomorphism such that σ(Fk ) consists entirely of polynomially growing automorphisms of Fn then the group Fn oσ Fk acts properly isometrically on some finite dimensional CAT(0) cube complex, and in particular is a-T-menable in the sense of Gromov. 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 [4]). In order to get the above statement, we prove the existence of a “space with walls” structure as introduced by Haglund and Paulin [12]. A theorem of Chatterji-Niblo [6] or Nica [15] (for similar constructions in other settings, see also [14], [16] or [10]) gives the announced action on a CAT(0) cube complex. A semi-direct structure Fn oσ Fk only depends on the class of σ(Fk ) in the group of outer automorphisms of Fn , denoted by Out(Fn ). Let us recall that Out(Fn ) = Aut(Fn )/Inn(Fn ), where Inn(Fn ) is the group of inner automorphisms: αw (x) = w−1 xw for any x ∈ Fn . Thus the statement of the theorem engulfs the case of direct products which is straightforward and well-known. The interesting case of the theorem is when we have a monomorphism σ : Fk ,→ Out(Fn ). The proof of Theorem 1 heavily relies upon the profound structure theorem of [3] about subgroups of polynomially growing automorphisms. We only appeal to the most elementary results of this theory. The important feature for us is the existence of an invariant tree, or equivalently of a particular graph-representative termed “improved relative train-track map”, for the whole subgroup of automorphisms: such an invariant 2

tree does not exist for non-abelian free subgroups containing non-polynomially (that is exponentially) growing automorphisms. 1. An invariant tree for unipotent subgroups of Out(Fn ) Definition 1.1 ([3]). An outer automorphism of Fn is unipotent if the automorphism that it induces on H1 (Fn ; Z) = Zn is unipotent. Lemma 1.2 ([2],Corollary 5.7.6). Any subgroup of polynomially growing outer automorphisms of Out(Fn ) admits a unipotent subgroup of finite index. Definition 1.3 ([3]). A filtered graph of Fn is a graph Γ with fundamental group isomorphic to Fn equipped with a filtration ∅ = Γ0 ( Γ1 ( · · · ( Γi ( Γi+1 ( · · · ( Γr = Γ where for each i = 0, · · · , r − 1, Γi+1 is the union of Γi with an edge Ei . A homotopy equivalence f of a filtered graph Γ is filtered if for any edge Ei of Γ, i = 0, · · · r − 1, f (Ei ) = vi Ei ui where vi and ui are loops contained in Γi . Observe that a filtered homotopy equivalence of a filtered graph Γ of Fn fixes each vertex of Γ. We assume fixed an identification of the fundamental group of Γ with Fn , which is equivalent to fix a homotopy equivalence between Γ and the rose with n petals (whose petals are identified to the generators of Fn ). A filtered homotopy equivalence naturally defines in this way an outer automorphism of Fn . Once chosen a base-point, it naturally defines an automorphism of Fn . The set of all filtered homotopy equivalences up to homotopy relative to the vertices of Γ, equipped with the composition, defines a group ([3][Lemma 6.1]) that we denote by F. Any choice of k filtered homotopy equivalences of a filtered graph Γ of Fn defines a subgroup hf1 , · · · , fr i of F and thus a subgroup U of Out(Fn ). When given a graph Γ, we denote by V (Γ) its set of vertices. The edges will always assumed to be equipped with some orientation: we denote by E + (Γ) the set of positively oriented edges, that is the edges equipped with this orientation, by E − (Γ) the set of negatively oriented edges, that is the edges equipped with the opposite orientation and then we set E(Γ) := E + (Γ) ∪ E − (Γ). Theorem 1.4 ([3]). For any integers n ≥ 2 and k ≥ 1, for any k-generated unipotent subgroup U of Out(Fn ), there exists a filtered graph Γ of Fn , with Card(E + (Γ)) ≤ 3n − 1, and k filtered 2 homotopy equivalences of Γ defining U. Let us recall that a k-generated group is a group which admits a generating set S with k elements. Corollary 1.5. For any integers n ≥ 2 and k ≥ 1, for any k-generated unipotent subgroup U of Out(Fn ), there exists a filtered graph Γ of Fn with Card(E + (Γ)) ≤ 3n − 2 and k filtered homotopy equivalences f1 , · · · , fk of Γ defining U which satisfy, for any positively oriented edge Ei of Γ and any j = 1, · · · , k, fj (Ei ) = Ei ui,j where ui,j is a loop in the graph Γi of the given filtration of Γ. Proof. Let Ei be any edge of Γ such that for some filtered homotopy equivalence fj , fj (Ei ) = ui,j Ei vi,j with ui,j non-empty. Subdivide Ei at its midpoint and orient the two resulting edges from the midpoint to the vertices of Ei . A continuous map from an 3

interval onto itself has a fixed point. By a homotopy of fj supported in a compact subset of Ei , one gets a map which admits the midpoint as fixed point. Doing so for each such couple map - filtered homotopy equivalence (Ei , fj ) gives the corollary.  2. Space with walls structure Spaces with walls were introduced in [12] in order to check the Haagerup property. 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 group 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. Theorem 2.1 ([12]). A discrete group G which acts properly on a space with walls is a-T-menable. In order to get Theorem 1 we will need the (stronger) result below (we refer to [15] for a similar statement). Let (X, W) be a space with walls. 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 nonempty. We denote by I(W) the (possibly infinite) supremum of the cardinalities of finite collections of walls which pairwise cross. Theorem 2.2 ([6]). Let G be a discrete group which acts properly on a space with walls (X, W). Then G acts properly isometrically on some I(W)-dimensional CAT(0) cube complex. In particular it is a-T-menable. 3. Preliminaries and notations 3.1. Two basic lemmas. Lemma 3.1. Given any two groups G and H together with a monomorphism σ : G ,→ Aut(H), if G0 is a finite index subgroup of G then H oσ G0 is a finite index subgroup of H o G. Proof. Any element of H oσ G is uniquely written as gh, g ∈ G and h ∈ H. Since G0 has finite index in G, there exists a finite number of elements g1 , · · · , gr such that any g ∈ G is in some left-class gi G0 . Then for any element gh of H oσ G, there is some i such that gh ∈ gi G0 h. Any element in G0 h is in the semi-direct product H oσ G0 hence the conclusion.  Lemma 3.2. If G is a group which admits an a-T-menable finite-index subgroup G0 then G is a-Tmenable. Proof. Let f be a proper conditionally negative function on G0 . There are a finite number of elements g1 , · · · , gr such that for any g ∈ G there is a unique gi such that g ∈ gi G0 . Defining fb(g) by f (gi−1 g) yields a proper conditionally negative function fb on G.  3.2. The group G and its horizontal and vertical subgroups. We denote by G = Fn oσ Fk the semi-direct product of the rank n and rank k free groups with U = σ(Fk ) a unipotent subgroup of Out(Fn ). We set Fn = hx1 , · · · , xn i, Fk = ht1 , · · · , tk i and αi = σ(ti ) for i = 1, · · · , k. We denote by αi , i = 1, · · · , r, r automorphisms of Fn such that hαi i = hαi i/Inn(Fn ) and U = hα1 , · · · , αr i/Inn(Fn ). 4

The group G admits as generating set S = {x1 , · · · , xn , t1 , · · · , tk } and as relators = αj (xi ) for i = 1, · · · , n and j = 1, · · · , k. We will term horizontal subgroup the normal subgroup Fn = hx1 , · · · , xn i and vertical subgroup the subgroup Fk = ht1 , · · · , tk i. 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. A reduced word in S ∪ S −1 is a word without any cancellation xx−1 or x−1 x (x ∈ S). Words consisting of vertical (resp. horizontal) letters are vertical (resp. horizontal) words. A reduced representative of an element g in G is a reduced word in S ∪ S −1 which defines g. t−1 j xi tj

3.3. The filtered graph Γ and its universal covering. Corollary 1.5 gives a filtered graph Γ, with filtration ∅ = Γ0 ( Γ1 ( · · · ( Γi ( Γi+1 ( · · · ( Γr = Γ and r filtered homotopy equivalences f1 , · · · , fr with fi inducing αi as an outer automorphism of Fn (in particular hα1 , · · · , αr i = U). We may assume that, once chosen a base-point, each fi induces the automorphism αi . We denote by Ei the positively oriented edges of Γ and write fj (Ei ) = Ei ui,j as in the statement of Corollary 1.5. Each positively oriented edge of Γ either connects a valency 2-vertex v0 to a valency m-vertex v1 (m ≥ 3), and is oriented from v0 to v1 , or is a loop. We denote by πΓ : T → Γ the universal covering of Γ: it comes equipped with a free, cocompact, isometric left-action of Fn , the free group being the group of decktransformations of this covering. Let us consider a maximal tree in Γ, where a maximal tree here denotes a tree containing all the vertices of Γ. We denote by T ⊂ T a lift of this maximal tree in T and by T c the edges of T incident to vertices in T (they are lifts of the complement of the maximal tree in Γ). We also choose a valency m-vertex with m ≥ 3 as base-point O ∈ T . We get an identification of a subset of the vertices of T with Fn by looking at the orbit {w.O}w∈Fn . There is the same number of distinct Fn -orbits than the ei the oriented edge of T ∪ T c with πΓ (E ei ) = Ei . number of vertices in Γ. We denote by E c The other edges of T are Fn -translates of the edges of T ∪ T and are oriented so that the action of Fn preserves the orientation. However, in order to simplify the notations, we ei as a “label” in order to describe an edge-path in T , will also sometimes use the letter E this edge-path being then uniquely defined once indicated its initial or terminal vertex. Let us recall that an edge-path in a graph is a continuous path between two vertices of the graph which does not backtrack in the interior of the edges, i.e. is locally injective at the points whose images lie in the interior of the edges. There is a natural partial ordering on the edges of Γ: the edge Ei is higher than the edge Ej if i > j and for some integer l ≥ 1, fkl (Ei ) contains Ej . This partial ordering naturally extends to a partial ordering on the edges of T . An edge which is maximal with respect to this partial ordering is a topmost edge. In T , each edge in Fn .T c inherits a label xi from the generating set of Fn . We will sometimes refer to them as xi -edges. An edge-path from a vertex γ.e vj to a vertex µ.e vj l 1 reads a word in the xi -edges of the form xi1 · · · xil , j = ±1, which defines the element γ −1 µ. If the edge-path is reduced then the word in the x±1 also is, where an edge-path is i reduced if it is locally injective (in a tree there is an unique reduced edge-path between any two vertices). 3.4. The mapping-cylinder. We denote by fei : T → T a lift of fi , i.e. πΓ ◦ fei = ej ) = E ej u fi ◦ πΓ , with fei (E ei,j (πΓ (e ui,j ) = ui,j ). We consider copies of T indexed over the elements of the vertical subgroup Fk , denoted by Tt . The mapping-cylinder C of T under 5

{fe1 , · · · , fek } is the space obtained from

G

Tt × [0, 1] by identifying x ∈ Tt × {1} with

t∈Fk

fej (x) ∈ Tttj × {0} for any t ∈ Fk , any x ∈ Tt × {1} and any j = 1, · · · , k. This mappingcylinder C has a structure of 2-dimensionnal cell-complex, the 1-skeleton of which is denoted by K: this 1-skeleton is composed of all the Fk copies Tt of T together with all the edges ve × [0, 1], connecting ve × {0}, with ve any vertex of Tt × {0}, to fej (e v ), a vertex of Tttj × {0}. These last edges will be termed vertical edges, whereas the edges in the Tt are called horizontal edges. Let us recall that some of the horizontal edges carry a label −1 xj , termed xj -edges. Similarly vertical edges carry a label t+1 according to the j , or tj orientation one considers and they are called tj -edges. There is a copy of the base-point O ∈ T ⊂ T , which was chosen to identify a subset of the vertices of T with Fn , in the tree Te . Since there is a free, isometric, cocompact left-action of the group G on C, and on K, we choose this copy, still denoted by O, as base-point to identify the subset {g.O}g∈G of the vertices of K with G. There is a well-defined free right-action of the vertical subgroup Fk on the set of vertices V (K): indeed if each fej : T → T is only a continuous map, its restriction to V (T ), that we will denote by Vej , is a homeomorphism. This allows one to define the action as follows: for any vertex ve of K and for any vertical element t = tj11 · · · tjm ∈ Fk , m ve.t = (Vejmm ◦ · · · ◦ Vej11 )(e v ). ei tj u−1 E e −1 −1 Finally each 2-cell C is a rectangle whose boundary reads a word of the form E i,j i tj . There are thus exactly two vertical tj -edges in the boundary of C, of which they form ei tj ui,j whereas the vertical boundary. The top of C is the horizontal edge-path reading E ei in T ∪ T c . the bottom of C consists of a single edge which is a left-translate of some E Top and bottom form the horizontal boundary of C. 4. Vertical walls 4.1. Definition and stabilizers. Definition 4.1. 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 reduced representative. A vertical j-wall is a left-translate g(Vj , Vjc ), g ∈ G. The following lemma is straightforward. Lemma 4.2. The collection of all the vertical walls is G-invariant under the left-action of G on itself. The horizontal subgroup is the left- and right-stabilizer of any vertical j-wall. 4.2. Finiteness of vertical walls between any two elements. Proposition 4.3. 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 Fn 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 4.2 each vertical wall is stabilized by the right-action of the horizontal subgroup. Thus, whatever horizontal generator xi is considered, no vertical wall separates g from gxi . The proposition follows.  6

5. Horizontal walls 5.1. Definition and stabilizers. We recall that T denotes a lift under πΓ of a maximal tree in Γ, whereas T c denotes the edges in T \ T which are incident to vertices of T (they are lifts under πΓ of the edges in the complement in Γ of the chosen maximal tree). We identify T and T c to their copies containing the base-point O in Te . Definition 5.1. ei be any topmost edge of T ∪ T c . The horizontal E ei -block Hi is the subset of all Let E ei w the elements of G which are the terminal vertex of a reduced edge-path of the form tE e where t is any vertical edge-path starting at the initial vertex of Ei and w is a horizontal edge-path. ei -block. A horizontal wall is a left-translate g(Hi , Hic ), g ∈ G, of a horizontal E Lemma 5.2. The collection of all the horizontal walls is G-invariant under the left-action of G on itself. The vertical subgroup is the left- and right-stabilizer of any horizontal wall. Proof. The first assertion is obvious since by definition all the horizontal walls are left G-translates of the horizontal blocks. Moreover, by definition, a horizontal block is leftinvariant under the action of the vertical subgroup, hence the assertion that the vertical subgroup is the left-stabilizer of any horizontal wall. Each vertical generator tj acts on the right by the map fj . By definition of a topmost edge Ei , fj sends Γ\{Ei±1 } to itself. It ei since E ei does not appear follows that the right-action of tj on Te preserves the sides of E e ±1 }). Hence the vertical subgroup also is the right-stabilizer of in the image of fej (T \ {E i any horizontal wall.  5.2. Finiteness of horizontal walls between any two elements. Proposition 5.3. There are a finite number of horizontal walls between any two elements in G. Proof. By the right-invariance under the action of the vertical subgroup (Lemma 5.2), it is sufficient to prove the proposition for any two horizontal elements g, h. Since a horizontal wall is crossed only when crossing an associated topmost edge, the number of horizontal walls between g and h is equal to the number of topmost edges in the edge-path in T from g to h. Hence the conclusion.  6. Vertizontal walls 6.1. Definition and stabilizers. Let us recall that th inversion of an edge is the map .−1 : E 7→ E −1 which to E ∈ E(Γ) assigns E −1 ∈ E(Γ). Definition 6.1. Let Er be a non-topmost edge of Γ. (1) The horizontal elementary Er -forbidden edges are, up to inversion: er in T ∪ T c . (a) The edge E (b) For any Ei 6= Er which is a loop and such that there is some fj such that fj (Ei ) = Ei ui,j contains Er , the horizontal edge with same label as Ei , and er . with origin the initial vertex of the edge-path reading ui,j and containing E er (g ∈ G) is the set The left-horizontal side (resp. right-horizontal side) of g.E of all vertices ve of Tg such that the unique reduced edge-path in Tg from ve to 7

er crosses an even number (resp. an odd number) of left the initial vertex of g.E g-translates of horizontal elementary Er -forbidden edges. (2) The vertical elementary Er -forbidden edges are, up to inversion, the vertical tj er and end on the other side of (E ei tj E e −1 ).E er . edges which start on one side of E i (3) The elementary Er -forbidden edges are the edges which are either horizontal or vertical elementary Er -forbidden edges. An elementary forbidden edge is an edge which is an elementary Er -forbidden edge for some edge Er of Γ. Remark 6.2. For a given non-topmost edge Er , the number of elementary Er -forbidden edges is bounded above by |E(Γ)|. Hence the total number of elementary forbidden edges is bounded above by |E(Γ)|(|E(Γ)| − 1). If one forgets the orientation of the edges, this bound becomes |E + (Γ)|(E + (Γ)| − 1). Definition 6.3. With the notations of Definition 6.1: (1) A vertical generator tj is Er -forbidden if and only if there exists an elementary forbidden vertical tj -edge. A horizontal generator xl is Er -forbidden if and only if the unique reduced loop based at πΓ (O) associated to xl contains an odd number of the images of Er -forbidden edges under πΓ . (2) The Er -authorized subgroup is the subgroup of G generated by: (a) The horizontal and vertical generators which are not Er -forbidden. (b) For any Ei 6= Er such that there is some fj such that fj (Ei ) contains Er , the ei tj E e −1 . element associated to the edge-path E i (c) For any Ei 6= Er which is a loop and such that there is some fj such that e −1 tj E ei . fj (Ei ) contains Er , the element associated to the edge-path E i The elements of the Er -authorized subgroup are the Er -authorized elements. (3) The (vertical, horizontal) Er -forbidden edges are the translates of the (vertical, horizontal) elementary Er -forbidden edges under the left-action of the Er authorized subgroup. A forbidden edge is an edge which is Er -forbidden for some edge Er of Γ. Definition 6.4. Let Ei be any non-topmost edge of Γ. The left-side (resp. right-side) of ei (g ∈ G) is the set of all vertices ve in K such that there exists an edge-path from ve g.E ei which crosses an even number (resp. an odd number) of left to the initial vertex of g.E g-translates of Ei -forbidden edges. In order to define the vertizontal walls, we need to prove the following result: Proposition 6.5. The left- and right sides of any non-topmost edge of K define a partition of V (K) in two disjoint, non-empty, classes. This proposition relies on the following easy lemma: Lemma 6.6. Let Er be a non-topmost edge of Γ.The boundary of any 2-cell of the mapping-cylinder reads a word with an even number of Er -forbidden edges in its boundary. 8

Proof. A vertical tj -edge is Er -forbidden if and only if there are g, h ∈ G such that the er it starts at differs from the horizontal side of (E ei tj E e −1 ).(g.E er ) it horizontal side of g.E i ends at. Assume that C is a 2-cell whose vertical edges are not Er -forbidden, or whose both vertical edges are Er -forbidden. Then the bottom of C is Er -forbidden if and only if the top is of C is Er -forbidden, hence an even number of Er -forbidden edges in the horizontal boundary of C. Since the vertical boundary also contains an even number of Er -forbidden edges, the conclusion follows in this case. If exactly one vertical edge in the boundary of C is Er -forbidden then the bottom of C is Er -forbidden if and only if the top is not. Hence an odd number of Er -forbidden edges in the horizontal boundary and the conclusion since the vertical boundary contains exactly one Er -forbidden edge.  Proof of Proposition 6.5. Let us consider some non-topmost edge Er of Γ. Let p be an edge-path in K. The homotopy in C, relative to the endpoints of p, from p to any other edge-path in K with the same endpoints, is a sequence of substitutions of subpaths of p by their complement in the boundary of some 2-cells. Since the boundary of each 2-cell contains an even number of Er -forbidden edges, each of this substitution preserves the parity of the number of Er -forbidden edges. Hence any two edge-paths in K which have the same endpoints have the same parity of Er -forbidden edges. It follows that the lefter define a partition of V (K) in two classes. The left-side contains by and right-sides of E er , hence is non-empty. Since the terminal vertex of E er is definition the initial vertex of E er , it is on the right-side, which is then connected to the initial one by the Er -forbidden E non-empty. Proposition 6.5 is proved.  Proposition 6.5 allows one to give the following Definition 6.7. er Let Er be a non-topmost edge of Γ and let Lr be the intersection of the left-side of E c with G. A vertizontal Er -wall is any left-translate g(Lr , Lr ), g ∈ G. The following lemma is obvious from the definition of the vertizontal walls and authorized subgroups: Lemma 6.8. The collection of all the vertizontal walls is G-invariant for the left-action of G on itself. The left-stabilizer of any vertizontal Er -wall is a conjugate of the Er -authorized subgroup. 6.2. Finiteness of the number of vertizontal walls between any two elements. Proposition 6.9. There are a finite number of vertizontal walls between any two elements in G. Proof. Let Er be a non-topmost edge of Γ and consider some vertizontal Er -wall g(Lr , Lcr ). e which is not the left g-translate of some elementary By definition, crossing an edge E Er -forbidden edge does not change the side of this vertizontal Er -wall. Since the leftstabilizer of an edge is trivial, once chosen an elementary forbidden edge x, there is at e Since there are only a finite number of most one element g ∈ G such that gx = E. elementary forbidden edges (see Remark 6.2), the number of vertizontal walls crossed when crossing any edge of K is finite.  7. A proper action Although obvious, the following proposition is indispensable: 9

Proposition 7.1. The set of all the horizontal, vertical and vertizontal walls defines a space with walls structure (G, W) for G. The left action of G on itself defines an action on this space with walls structure. Proof. By Propositions 4.3, 5.3 and 6.9 there are a finite number of walls between any two elements so that (G, W) is a space with walls structure.  We now prove the following result: Proposition 7.2. The action of G on the space with walls structure (G, W) given by Proposition 7.1 is proper. Proof. Consider a sequence of elements (gn ) ⊂ G which tends to infinity. Each gn is written tn hn where tn is a vertical element and hn a horizontal element. If the lengths of the tn ’s tends to infinity, then, since the vertical walls are the classical walls of the free group, the same is true for the number of vertical walls crossed by the gn ’s. This allows one to consider only sequences of horizontal elements (hn ) tending to infinity. Let (pn ) be the associated sequence of reduced edge-paths in T from e to hn . If the number of topmost edges crossed by the pn ’s goes to infinity then so does the number of horizontal walls crossed: indeed, like a classical wall of the free group, the horizontal wall associated to any topmost edge has exactly 2 sides, hence a horizontal wall will be crossed at each occurrence of a topmost edge in pn and never crossed back. We can thus assume that the pn ’s only contain non-topmost edges. The non-topmost horizontal forbidden edges associated to a given wall are all left-translates by authorized horizontal elements of a single forbidden elementary edge. Thus, if x is a forbidden edge, then there will be no other forbidden edge associated to the same wall in any geodesic in T starting at e and containing x. This readily implies that any pn contains at most one forbidden edge for each vertizontal wall. Hence the number of vertizontal walls associated to the non-topmost edges goes to infinity with their number in pn . The proof of Proposition 7.2 is complete.  8. The Haagerup property and dimension of the cube complex We give here, as corollaries of the construction developed above, the two main results we were interested in: the Haagerup property for G and the, stronger, fact that G acts properly isometrically on a cube complex whose dimension is bounded above by 3n−2+k (Theorem 1). Corollary 8.1. The group G satisfies the Haagerup property. Proof. By Propositions 7.1 and 7.2, G acts properly on a space with walls structure (G, W). By [12] G satisfies the Haagerup property.  Corollary 8.2. The group G acts properly isometrically on some finite dimensional cube complex, the dimension of which is bounded above by |E + (Γ)| + k ≤ 3n − 2 + k. Proof. By Corollary 8.1 and Theorem 2.2, the group G acts properly isometrically on a cube complex whose dimension is equal to the supremum of the cardinalities of the sets of walls which pairwise cross (see Section 2). Since vertical walls are classical walls of the free group Fk , there is at most one vertical wall in such a collection. By definition, a 10

horizontal wall consists of a collection of classical walls of the free group Fn , one for each vertical translate of Fn which are all copies of a same wall of Fn . It readily follows that a set of walls which pairwise cross also contains at most one horizontal wall associated to a given topmost edge. Given two distinct topmost edges, as before, one connected component of T cut by one is contained in a connected component cut by the other. Thus a set of walls which pairwise cross contains at most one horizontal wall. Let us now consider two distinct vertizontal Er -walls (Lr , Lcr ) and g(Lr , Lcr ). Since the Er -authorized subgroup is the left-stabilizer of (Lr , Lcr ), g is a Er -forbidden element. Thus, for any vertical element t, the horizontal forbidden edges of g(Lr , Lcr ) are all contained in some connected component of Tt deprived of the horizontal forbidden edges of (Lr , Lcr ), and all these connected components are copies of a same one in Te . Hence, for each non-topmost edge Er , there is at most one Er -vertizontal wall in a set of walls which pairwise cross. It follows that the cardinality of a set of walls which pairwise cross is bounded above by the number of non-topmost edges plus one (a horizontal wall) plus k (the rank of the vertical subgroup). In order to count the vertizontal walls, one counts the non-topmost oriented edges. The number of non-topmost oriented edges is at most |E + (Γ)| − 1, hence the conclusion (see Corollary 1.5 for the upper-bound depending only on n and k).  Proof of Theorem 1. We consider the group Fn oσ Fk with σ(Fk ) a subgroup of Out(Fn ) which consists entirely of polynomially growing automorphisms. By Lemma 1.2, Fk admits a finite-index subgroup U such that σ(U) is a unipotent subgroup of Out(Fn ). By Schreir’s theorem, U is a free group and since it is of finite-index in Fk , it is a finitely generated subgroup. Let us set G := Fn oσ U. By Corollary 8.1, G satisfies the Haagerup property. By Lemma 3.1, since U has finite-index in Fk , G is of finite-index in Fn oσ Fk . Thus, by Lemma 3.2, Fn oσ Fk satisfies the Haagerup property. By Corollary 8.2, G acts properly isometrically on some finite dimensional cube complex R. We get an isometric action of Fn oσ Fk on R by defining g.x := (gi−1 g).x if g ∈ gi G. Since G is of finite-index in Fn oσ Fk , there are a finite number of left G-classes. This implies that this action of Fn oσ Fk on R is proper since the action of G was.  References [1] Charles A. Akemann and Martin E. Walter. Unbounded negative definite functions. Canadian Journal of Mathematics. Journal Canadien de Math´ematiques, 33(4):862–871, 1981. [2] Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(Fn ). I. Dynamics of exponentially growing automorphisms. Annals of Mathematics, 151(2):517–623, 2000. [3] 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. [4] 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. [5] M. Burger. Kazhdan constants for SL(3, Z). Journal f¨ ur die Reine und Angewandte Mathematik, 413:36–67, 1991. [6] 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. [7] 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. [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] F. Gautero. A non-trivial example of a free-by-free group with the haagerup property. Groups, Geometry and Dynamics, 2012. 11

[10] Erik Guentner and Nigel Higson. Weak amenability of CAT(0)-cubical groups. Geometriae Dedicata, 148:137–156, 2010. [11] Uffe Haagerup. An example of a nonnuclear C ∗ -algebra, which has the metric approximation property. Inventiones Mathematicae, 50(3):279–293, 1978/79. [12] 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. [13] Paul Jolissaint. Borel cocycles, approximation properties and relative property T. Ergodic Theory and Dynamical Systems, 20(2):483–499, 2000. [14] Graham A. Niblo and Martin A. Roller. Groups acting on cubes and Kazhdan’s property (T). Proceedings of the American Mathematical Society, 126(3):693–699, 1998. [15] Bogdan Nica. Cubulating spaces with walls. Algebraic & Geometric Topology, 4:297–309 (electronic), 2004. [16] Michah Sageev. Ends of group pairs and non-positively curved cube complexes. Proceedings of the London Mathematical Society, 71(3):585–617, 1995. ´ de Nice Sophia Antipolis, Laboratoire de Mathe ´matiques Franc ¸ ois Gautero, Universite ´ J.A. Dieudonne (UMR CNRS 7351), Parc Valrose, 06108 Nice Cedex 2, France E-mail address: [email protected]

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