Proper actions on $\ell^p$ spaces for relatively hyperbolic groups

We show that for any group $G$ that is hyperbolic relative to subgroups that admit a proper affine isometric action on a uniformly convex Banach space, then $G$ acts properly on a uniformly convex Banach space as well.


Introduction
Kazhdan's property (T) was introduced in 1967 by D. Kazhdan in [K] and since then has been intensively studied. A reformulation of Kazhdan property (T), following from work of Delorme-Guichardet, is that any action on a Hilbert space has a fixed point. Groups with property (T) include higher rank lattices in simple Lie groups, such as SL(n, Z) for n ≥ 3, but also some lattices in rank one Lie groups, such as lattices in Sp(n, 1) for n ≥ 3. The uniform lattices in Sp(n, 1) being hyperbolic, it shows that some hyperbolic groups can have property (T), as opposed to some others, like free groups or surface groups, that admit a proper action on a Hilbert space. The latter is called Haagerup property or aTmenability. The first natural generalization of a Hilbert space are L p space and it is implicit in P. Pansu's results in [Pa], that lattices in Sp(n, 1) admit an action without fix points on an L p space. G. Yu in [Y] shows that in fact any hyperbolic group admits a proper action on an p -space for p large enough. M. Bourdon in [Bou], B. Nica in [N] and more recently A. Alvarez and V. Lafforgue in [AL] gave an alternative proof of Yu's theorem. On the contrary, higher rank lattices are known to not act on any L p or uniformly convex Banach space ( [BFGM] and [L]).
Relatively hyperbolic groups are a geometric generalization of hyperbolic groups and have been studied a lot. They allow interesting group constructions while retaining a lot of the geometry of hyperbolic spaces. Relatively hyperbolic groups can also have property (T), but their peripheral subgroups can forbid any action on a uniformly convex Banach space. We show that this is in fact the only obstruction. Working with Alvarez-Lafforgue's construction, we prove that any relatively hyperbolic group has a non-trivial affine isometric action on an p space, for some p large enough. If furthermore its peripheral subgroups act properly by affine isometries on some uniformly convex Banach space, then the whole group also acts properly on a uniformly convex Banach space. It is the case for instance for Haagerup peripheral subgroups, in particular for all amenable ones. Our main result is the following.
Theorem 0.1. Let G be a relatively hyperbolic group, non-elementary. Then, for p sufficiently large G admits an isometric action on an p -space with an unbounded orbit.
MSRI during the special semester in Geometric Group Theory, and continued during the INI special semester Non-positive Curvature group actions and cohomology, supported by EPSRC grant no EP/K032208/1. The authors would like to thank both programs for the great working conditions. Both authors are partially supported by IUF, and the first author by ANR GAMME.

Working in relatively hyperbolic groups
To fix notations we recall that a graph X is a set X (0) , the vertex set with a set X (1) ⊆ X (0) ×X (0) , the oriented edges, endowed with an origin map o : X (1) → X (0) and a fixed-point free involution reversing the edges¯: X (1) → X (1) . A pair {e,ē} is an unoriented edge. We denote by t : X (1) → X (0) the compositionō, this is the terminus map. We assume the reader familiar with paths, connectedness, length, geodesics, in graphs, as well with hyperbolicity. All our graphs are considered with their graph metric where an edge has length one.
Definition 1.1. Let X be a graph. Given a vertex v and two oriented edges e 1 , e 2 such that o(e 2 ) = t(e 1 ) = v, the angle between e 1 and e 2 at v, denoted by v (e 1 , e 2 ), is the infimum (in R + ∪ {+∞}) of length of paths from o(e 1 ) to t(e 2 ) that avoid v. In other words, v (e 1 , e 2 ) = d X\{v} (o(e 1 ), t(e 2 )). By convention, we will use the following abuse of notation: if x 1 , x 2 , c are vertices in X, we say that c (x 1 , x 2 ) > θ if there exists two edges e 1 , e 2 with t(e 1 ) = o(e 2 ) = c, with e i on a geodesic between c and x i (i = 1, 2) and such that c (e 1 , e 2 ) > θ. We say that c (x 1 , x 2 ) ≤ θ otherwise, namely, if for any geodesic between c and x i and starting by and edge e i (i = 1, 2), then c (e 1 , e 2 ) ≤ θ.
Large angles at a vertex in a δ-hyperbolic graph will force geodesic through that vertex. More precisely we have the following.
(1) If c a, b > 12δ, then all geodesics from a to b contain c.
(2) If e, e are two edges originating at c and that are on geodesics from c to a, then c (e,ē ) ≤ 6δ. (3) For any θ ≥ 0, if c (x, y) > θ + 12δ then for all choice of edges 1 , 2 at c starting geodesics respectively to x and to y, c ( 1 ,¯ 2 ) ≥ θ. Proof.
(1) Applying δ-thiness of triangles on the geodesics [c, a] and [c, b] at distance 2δ from c, we obtain a path of length at most 12δ, containing an arc of [a, b], and that goes from the end of the first edge of [c, a] to the end of the first edge of the [c, b]. If the angle at c is larger than 12δ this path cannot avoid c. But only the arc on [a, b] can meet c again. Hence the first claim.
(2) Any geodesic bigon between c and a must be δ-thin, so if we take a geodesic bigon starting with the edges e and e and look at the points at distance 2δ from c, we get a path of length 6δ at most and that avoids c.
(3) This is a consequence of (1) and (2), combined with the triangular inequality for angles. This definition of fine graphs is equivalent to Bowditch's definition that, for all edge, for all number, there are only finitely many simple loops of this length passing through this edge, [Bow].
Definition 1.4. Let G be a finitely generated group, and H 1 , . . . , H k , subgroups of G. Consider a Cayley graph CayG of G (over a finite generating set), and construct the cone-off over H 1 , . . . , H k as follows: for all i and all left coset gH i of H i , add a vertex gH i and for each h ∈ H i , add an edge between gH i and gh. We denote by Cay(G) this graph, called a coned-off Cayley graph (of G with respect to the subgroups H 1 , . . . , H k ).
We can now recall the definition of Bowditch of relative hyperbolicity from [Bow] (his original definition is about G-graphs, but this is an equivalent formulation from [D03,Appendix] A difficulty of working in relatively hyperbolic groups is the lack of local finiteness in the coned-off graph. The angles, and below the cones, are useful tools for working around this. The following estimate says that the word metric is equivalent to the coned-off metric to which we add the sum of the angles at infinite valence vertices. We will be needing the first inequality only. Proposition 1.6. Let G be a group hyperbolic relative to {H 1 , . . . , H k } with d w a word metric, for a finite generating set containing generating sets of each H i . Let Cay(G) be its coned-off Cayley graph, for its relatively hyperbolic structure, withd the graph metric.
There are A, B > 1 for which, for all g ∈ G, for each choice of geodesic [1, g] in the coned-off graph Cay(G), if ([1, g]) denotes the sum of angles of edges of [1, g] at vertices of infinite valence, then Proof. First given a geodesic [1, g] in the coned-off graph Cay(G), we construct a path in the Cayley graph (whose length will bounded from above d w (1, g)) by replacing each pair of consecutive edges e, e at an infinite valence vertex, by a path in the corresponding coset xH i of H i . We remark that we can choose this path of length bounded above in terms of the angle, say θ. Indeed, by definion of angle, there exists a path in Cay(G) of length at most θ avoiding xH i , from o(e) to t(e ). Project each vertex in this path on the 1-neighborhood of xH i . One gets a sequence of θ points y 1 , . . . y θ in xH i , from o(e) to t(e ), two consecutive points being at an angle of at most 2δ at xH i . It follows that there are only boundedly many possible transitions y −1 i y i+1 in H i , and we have our bound on the word length in H i in term of the angle θ at xH i . Thus 1 For the second inequality, consider a geodesic from 1 to g in the word metric. It produces an injective continuous path q in Cay(G), that fellow-travels a geodesic [1, g] in the coned-off graph (see for instance [DS,Proposition 8.25]). The path q does not contain any vertex of infinite valence, therefore each time [1, g] contains a vertex of infinite valence, we can use the proximity of [1, g] with q and the fact that q does not contain this point, to estimate the angles in terms of the length of a subsegment of q. By thinness of the graph, each subsegment of q is only used at most a uniformly bounded amount of times. Therefore, the total sum of the angles at infinite valence points is at most a certain multiple of the length of q, which is the word length of g.
1.1. Cones. The notion of angles will now allow us to talk about cones in any graph X. Cones are a useful tool for working in relatively hyperbolic groups as they allow to take neighborhoods of geodesics in a finite way.
Definition 1.7. Let X be a graph. Given an oriented edge e and a number θ > 0, the cone of parameter θ around the oriented edge e, denoted Cone θ (e) is the subset of vertices v and edges of X such that there is a path from o(e), starting by e, and containing v or , that is of length at most θ and for which two consecutive edges make an angle at most θ.
For vertices of finite valence we can define the cone at a vertex of parameter θ by the union of all the cones of parameter θ around the edges adjacent to the vertex.
If angles inside a cone of parameter θ are by definition bounded by θ, angles at vertices close to the cone are also controlled in terms of θ. Precisely we have the following estimate.
Proof. Consider c with two edges 1 , 2 starting at c, on geodesics g 1 , g 2 toward o(e), and v respectively, of length ≤ θ/10. We also are given a path p starting by e, going to v of length at most θ, and maximal angle at most θ between consecutive edges. Concatenating the paths g 1 , p,ḡ 2 (orientation reversed), we have a loop at c, starting by 1 , and ending by 2 and of length at most 6θ/5. If this loop doesn't pass by c (except starting and ending point), then by definition of angle we have that c ( 1 ,¯ 2 ) ≤ 6θ/5, which would prove the claim. If this path passes at least twice at c, only the segment p can use c, and it can do so at most (θ − d(o(e), c) − d(c, v))/2 times, each time realizing an angle between incident and exiting edge of at most θ. By the triangular inequality on angles, this path gives a bound on the angle c ( 1 ,¯ 2 ) of at most θ(θ−d(o(e), c)−d(c, v))/2+6θ/5 ≤ (θ 2 +3θ)/2. This establishes the claim.
Cones also can be composed in an obvious way. Proposition 1.9. Let α, β ≥ 0, if e, e , e are three edges of a graph and if e ∈ Cone α (e ) and e ∈ Cone β (e) then e ∈ Cone α+β (e).
Proof. We have a path of length and maximal angle bounded by β that starts by e and ends by e orē , and another of length and maximal angle at most β that starts at e and contains e orē . Concatenate the two paths, possibly overlapping at e , or possibly cancelling a backtrack e ē , produces a path of length at most α + β and maximal angle at most max{α, β}.
Finally, the following proposition shows that cones are well-suited to work in hyperbolic and fine graphs.
In any δ-hyperbolic graph, geodesic triangles are conically thin: for any geodesic triangle, any edge on any side of the triangle is contained in a cone of parameter 50δ around a certain edges of one of the two other sides.
The above proposition allows us to deduce that intervals are finite in coned-off graphs, even if those graphs are not locally finite.
Proposition 1.11. In any hyperbolic fine graph, between any pair of points a, x the set of points in a geodesic between a and x is finite.
A more important use of cones was explored in [D06], and follows from the fact that in a hyperbolic graph, quasi-geodesics "without detours" are conically close to geodesic ( [D06,Prop. 1.11]). We will use a variant of it, exposed in the next section.
1.2. Conical α-geodesics. In order to define a flow in the next section, we need to be able to travel coarsely from a point toward another point in a uniformly finite way. In [AL], the authors use the concept of α-geodesic, for some α ≥ 0. Precisely, in a metric space, given two points x and a, we say that t belongs to an α-geodesic between x and a if d(x, t) + d(t, a) ≤ d(x, a) + α. Note that being on a 0-geodesic is being on a geodesic. However, when the space is not uniformly locally finite, the set of points in a α-geodesic between x and a can be infinite in general, if α ≥ 1, even if we assume the graph to be fine. We will use a variant of the notion of α-geodesics, that takes angles into account. Constants are not really relevant, but they need to be fixed by convention.
Definition 1.12. Let X be a graph, and x, a ∈ X (0) . For ρ ≥ 0 we denote by E a,x (ρ) the set of edges e with d(a, o(e)) = ρ that are contained in geodesics from a to x, or from x to a. We say that t ∈ X (0) belongs to an α-conical-geodesic between x and a if t is on an α-geodesic, d(a, t) ≤ d(a, x), and t ∈ e∈Ea,x(d(a,t)) Cone 40α (e).
We will denote U α [a, x] the set of points belonging to an α-conical-geodesic between x and a.
For each ρ ≥ 0, we denote by S a,x (ρ) the slice of U α [a, x] at distance ρ from a, that is the set We will now see some useful properties, namely that these sets are finite, stable (analogue of [AL,3.3] and [AL,3.4]), non-empty and slices at large angles are reduced to a point.
Proposition 1.13. Let X be a δ-hyperbolic fine graph, and let a, x ∈ X (1) (Finiteness) For any α > 0 the set U α [a, x] is finite. Each slice is contained in a cone of parameter 40α. ( (3) For all v on a geodesic between a and x, the slice of U 2δ [a, x] at distance d(a, v) from a contains v, hence is non empty. Proof.
(1) By definition U α [a, x] is contained in an intersection of cones, which are finite according to Proposition 1.10. Each slice, by definition, is contained in U α [a, x], which is an intersection of a family of cones of parameter 40α.
( [AL,3.4] Consider y ∈ [a, u] such that d(a, y) = d(a, v), and an edge e of [a, u] with origin y. Then v ∈ Cone 80δ (e).
Moreover, by conical thinness of triangles (Proposition 1.10), for an edge e of [a, x] at same distance from a, e ∈ Cone 50δ (e ). It follows, by Proposition 1.9, that v ∈ Cone 130δ (e ), which establishes the claim.
(4) Consider two edges e, e at c (i.e o(e) = o(e ) = c) on a geodesic [a, x], where t(e) is closer to a and t(e ) is closer to a. By Proposition 1.2, c (ē, e ) > (999δ) 2 . It follows, by Lemma 1.8, that Cone 80δ (ē) ∩ Cone 80δ (e ) = {c}. Thus, only c can be in the slice. Since the slice is non-empty, we have the lemma.

The flow on the coned-off graph
Let us recall that, given a set Y and an additive semigroup V , a flow is a map F : Y ×V → Y such that for all y, F (y, 0) = y and F (y, t 1 +t 2 ) = F (F (y, t 1 ), t 2 ). In this definition, the semi-group V is usually R + , but we will use N (by convention 0 ∈ N), and in this case the flow is entirely determined by the flow step, which is the map T : Y → Y defined by T (y) = F (y, 1).
In the following, X will be a δ-hyperbolic fine graph (for some integer δ > 0), and Proba(X) is the space of probability measures supported on X (0) . We will define a flow for the semi-group N, on the set Y = Proba(X) × X (0) , which is equivariant by the group of isometries of X. We think of it as a version of a geodesic flow.
When X is countable (which is the case in our context) elements of Proba(X) are just functions from X (0) to R + of sum 1, but we find it convenient and enlightening to see them as probability measures.
2.1. The flow step. In this subsection we will define the flow step First let us introduce some vocabulary. Given a and x, we say that the step T a at x is • ending if a is of infinite valence and 1 < d(a, x) ≤ 5δ, or if a has finite valence, d(a, x) ≤ 5δ and there exists c such that c (a, x) > 1000δ, • stationary in all other cases , i.e in any of the following three cases x = a, a is of infinite valence and x is a neighbor a, a is of finite valence, d(a, x) ≤ 5δ and for all c, c (a, x) ≤ 1000δ.
We denote by δ x the Dirac mass at x and define the flow step T by defining T a (δ x ), for a, x two vertices of X and then extend by linearity on probability measures: Definition 2.1. Let X be a δ-hyperbolic fine graph, and let a, x ∈ X (0) . We set r a,x to be the largest value of r such that 5δ r < d(a, x).
• If the step T a at x is either initial or regular, then T a (δ x ) is the uniform probability measure supported by the slice S a,x = S a,x (5δ r a,x ) (as in Definition 1.12). • If the step T a at x is ending, and a has infinite valence then T a (δ x ) is the uniform probability measure supported by the slice S a,x = S a,x (1) at distance 1 from a. If the step is ending and a has finite valence, then T a (δ x ) is the Dirac mass supported on the unique c minimizing d(a, c) such that Let us check that this definition makes sense. First, the set S a,x is well defined: in the case of an ending step, any vertex c such that c (a, x) > 900δ is on every geodesic between a and x, thus there is a unique one that is closest to a. Second, the set S a,x is never empty by Proposition 1.13(3). Third, it is always finite, by Proposition 1.13(1). Observe that among iterates of the flow step applied at (δ x , a), only the first iteration can be initial, only one step among the iterates can be ending, and after the ending step, all steps are stationary.
This defines uniquely the value of T a (δ x ) for all x and all a and so completely defines the flow step T , hence also the flow 2.2. Stationarity and mask. For two points x and a in X (0) , the flow step T a (δ x ) is a probability measure whose support is closer to a than δ x , so the flow step has pushed the point x towards a. Iterating this flow step will eventually stall and give a measure that is close to a.
Proposition 2.2. For X a δ-hyperbolic fine graph and all a, x ∈ X (0) , the flow step from Definition 2.1 is stationary in k: if k > r a,x +1, one has T ra,x+1 a In case of equality, then T a (x) = δ x . Indeed, if the flow step is initial or regular, then d(a, y) = d(a, y ) < d(a, x), since by definition, 5δ r a,x < d(a, x). The ending step case, also induces a strict inequality, and the stationary step case induces the equality T a (x) = δ x . Hence the flow eventually become stationary for k large enough.
Definition 2.3. Given any two vertices a and x of X a δ-hyperbolic fine graph, the mask of a for x is the probability measure given by where R(a, x) the minimal number of iterations of T a on δ x so that the stationary step is reached.
Let us observe the following.
Proposition 2.4. Let X be a δ-hyperbolic and fine graph. For a, x ∈ X (0) , then The last estimate is very crude, and with some closer look, one is likely to get a much more precise control. Proof.
(1) If a has infinite valence, by definition of the ending step in this case the measure is in the slice at distance 1 from a. By Proposition 1.13(2), the last slice used is in a 4δ-conical geodesic from a to x. So, given e which is first edge on a geodesic from a to x, by the definition of slices, this last slice used is contained in Cone 160δ (e).
(2) In the case of a finite valence vertex a, the condition for the step being stationary is that there are no angle larger than 1000δ between a and a point in the support. Again, by Proposition 1.13(2), the support is contained in a slice is in a 4δ-conical geodesic from a to x, hence in a cone Cone 160δ (e 0 ) centered at an edge in some geodesic [a, x], at distance at most 5δ from a. We moreover know, by definition of the ending step, that there is no angle larger than 1000δ between a and o(e 0 ). Thus, by composition of cones, Proposition 1.9, the support is contained in Cone 1160δ (e) for an initial edge of a geodesic [a, x].
The definition of the flow step being in a purely metric way, as a consequence the masks are equivariant with respect to the isometry group.
Proposition 2.5. Let X be a δ-hyperbolic fine graph. The map that associates to a pair of points (x, a) the mask of a for x is equivariant with respect to the isometry group of X. More precisely, if φ is an isometry of X, one has φ * µ x (a) = µ φ(x) (φ(a)).
2.3. The flow stays focused. In this subsection we will be recording the behavior of the flow step at different stages Lemma 2.6. Let X be a δ-hyperbolic fine graph, and for a ∈ X (0) let T a denote the flow step as in Definition 2.1. Let x, x ∈ X (0) .
(1) If x is such that the k first iterates of T a are initial or regular steps, then, for any two points y, y in the support of T k a (δ x ), one has d(y, y ) ≤ 5δ. Moreover the support of T k a (δ x ) is contained in a cone of parameter 160δ.
(2) If x, x are neighbors, and 5kδ < d(a, x) ≤ d(a, x ) < 5(k + 1)δ, then the union of the supports of T a (δ x ) and T a (δ x ) has diameter at most 8δ + 1. a) and assume that for both of them, the flow step is regular. Then the union of the supports of both measures T a (δ x ) and T a (δ x ) has diameter at most 8δ. Moreover, the union of these supports is contained in a cone of parameter 170δ, and their intersection is not empty.
(4) If x, x are neighbors and both at at distance larger than 5δ from a, then there are two exponents , that are either 1 or 0, such that T a (δ x ) and T a (δ x ) have support on a sphere centered at a of radius 5kδ for some k, and the diameter of their union is at most 8δ + 2. Proof.
(2) By assumption the steps of the flow are initial. Let y 0 and y 0 be respectively on geodesics [x, a] and [x , a] at distance 5kδ from a. By δ-hyperbolicity, they are at distance less than 2δ + 1 from each other. Let y, y respectively be in the supports of T a (δ x ) and T a (δ x ). Apply the argument of part (1) using that y and y are on 2δ-geodesics (there is only one step of flow). One obtains that d(y, y 0 ) ≤ 3δ and d(y , y 0 ) ≤ 3δ, hence d(y, y ) ≤ 8δ + 1.
(3) Here x and x are not assumed neighbors, but at distance 5(k + 1)δ from a. Let y 0 and y 0 respectively be on geodesics [x, a] and [x , a] at distance 5kδ from a. By δ-hyperbolicity, they are at distance less than 2δ from each other and the same argument than part (2) gives a bound of 8δ on the union of the supports of T a (δ x ) and T a (δ x ) We now prove that the intersection of those supports is non-empty. Given two geodesics [a, x] and [a, x ], we denote by e, e the edges of these geodesics starting at distance (d(a, x) − 5δ) from a. By δ-hyperbolicity, the two segments stay 2δ-close for a length of at least δ after e and e . One deduces (as in the proof of Proposition 1.2) that e ∈ Cone 10δ (e ). Therefore, by composition of cones (Proposition 1.9) Cone 160δ (e) ⊆ Cone 170δ (e ). Moreover, o(e) ∈ U 2δ [a, x ] and hence by Proposition 1.13(3) the intersection of the supports is non-empty.
(4) There are several cases, depending whether the flow from x respectively from x needs an initial step or not, in other words, whether x, respectively x are at distance strictly greater than 5kδ or equal 5kδ from a.
If the first step of the flow is initial for both (δ x , a) and (δ x , a), then part (2) says that the union of the supports of T a (δ x ) and T a (δ x ) has diameter at most 8δ. The same is true according to part (3) if both first steps of the flow are regular.
Assume that d(a, x) = 5kδ and d(a, x ) > 5kδ. Then we compare δ x (i.e = 0) with T a (δ x ) (i.e = 1). According to part (1) the support of T a (δ x ) has diameter at most 5δ, and by Proposition 1.13(3) it contains a point at distance 1 from x , hence at distance less than 2 from x, the union of the supports of T a (δ x ) and T a (δ x ) has diameter at most 5δ + 2, hence the result.
2.4. Check points at large angles. The following proposition says that when the flow passes a large angle, this large angle acts as a check point: The flow could have started at this large angle, and still give the same result.
Proposition 2.7. Let (X, d) a δ-hyperbolic fine graph. Let a, x be vertices in X.
If there exists c ∈ [a, x] such that c (a, x) > (2000δ) 2 , then for any x at distance 1 from x, we have that µ x (a) = µ x (a) = µ c (a).
Proof. Define r(c) to be the largest r so that (r a,x − r)5δ ≥ d(a, c). First notice that for all r ≤ r(c), and all b in the support of T r a (δ x ), one has a, c), and by Proposition 1.13(3) there is b on the geodesic [a, x] realizing the maximal angle at c, that is in the support of T r a (x). Hence c (a, b ) > (2000δ) 2 . Since b and b are in a same cone of parameter 160δ (Proposition 1.13(1)), the maximal angle between then is at most 2(160δ) 2 (by Lemma 1.8). Thus, c (a, b) > (1987δ) 2 > (1910δ) 2 as claimed.
We now show that T Thus, all geodesics from a to b pass through c and make an angle at least (1900δ) 2 . It follows that the intersection of all cones of parameter 80δ centered at an edge of E a,b (d(a, c)) is reduced to {c} (by Proposition 1.13(4)), hence T a (δ b ) = δ c , which means that T r(c) a (δ x ) = δ c , and T r(c)+1 a (δ x ) = T a (δ c ) as claimed. When d(a, c) is not a multiple of 5δ, apply T a to all points b in the support of T r(c) a (δ x ). But since we saw that c (a, b) > (1910δ) 2 , all geodesics from a to b pass through c and make an angle at least (1900δ) 2 . The points on 2δ-geodesics from a to c are exactly the points on 2δ-geodesics from a to b that are at distance at most d(a, c) from a, and we have equality of the sets E a,b ((r a,x − r(c))5δ) = E a,c ((r a,x − r(c))5δ). Hence T a (δ b ) = T a (δ c ), and and T r(c)+1 a (δ x ) = T a (δ c ) again as claimed. Finally, if x = c and x at distance 1 from x, then, for r (c) defined similarily to r(c) for x we have The angle at c between a and x can be reduced by one unit, which is still well above the threshold to apply the previous argument. Iterating the flow step gives the conclusion of the proposition.
2.5. Confluence. We will need to control that, given a, and x, x close to each other, but "far" from a (in terms of distance or of angles), each of the regular flow steps from (δ x , a) and from (δ x , a), are uniformly close. To do that, we will be talking of confluence. For η, η two probability measures on X, denote by M (η, η ) the measure defined by We also define the symetric difference of η and η as Note that it is always a positive measure since η, η are probability measures.
Definition 2.8. Let β ≥ 0. We say that the flow step T a is β-confluent on where the norm η of a measure is defined as its 1 -norm η = x∈X |η(x)|.
Assume now that X is a fine δ-hyperbolic graph with a co-finite group action. There are only finitely many orbits of cones of given parameter. Hence, there exists C > 1 that is an upper bound on the cardinality of cones of parameter 160δ in X. In the following, we will assume that X has a co-finite group action, which will ensure the existence of such C.
Proposition 2.10. Let X be a fine δ-hyperbolic graph with a co-finite group action. Let k ≥ 1, and η, η be two measures whose supports are on S(a, 5(k + 1)δ), and have union of diameter less than 8δ.
Then, the k consecutive flow steps on (η, a) and (η , a) are 1 C -confluent. In particular, for all m ≥ k, Proof. We proceed by induction to prove that for each i ≤ k, the union of supports of the measures T i a (η) and T i a (η ) have diameter at most 8δ. There is nothing to prove for i = 0. For i > 0, assuming T i−1 a (η) and T i−1 a (η ) have support whose union has diameter at most 8δ, we can say that, since all steps of the flow so far were regular, these supports are on the sphere S(a, 5(k + 2 − i)δ). We can apply Lemma 2.6(3) to obtain that the union of supports of T i a (η) and T i a (η ) have diameter at most 8δ.
We then prove the confluence. For each Dirac masses λ v δ v and λ v δ v , respectively in the support of T (η )), the regular step of the flow is 1 C -confluent by Lemma 3 (second part). Combining this confluence on all masses in these supports gives that Proposition 2.11. Let X be a hyperbolic fine graph, with a cofinite group action. There exists a constant κ < 1 such that, for all a, for all x 1 , x 2 at distance 1 from each other, if Θ denotes the minimal sum of angles at infinite valence vertices on a geodesic [a, x], then for d(a, x) + Θ sufficiently large, one has Observe that the statement would hold also for Θ defined as the sum of all consecutive angles, and also if it was taken to be the maximum of these sums.
2.6. Non-confluence. In this subsection we will be showing that for two given sources of the flow, any point on a geodesic between those two sources have disjoint masks for each of those sources.
Proposition 2.12. Let X be a δ-hyperbolic and fine graph, and let µ be the mask of Definition 2.3.
(1) If d(x, x ) ≥ 10δ and if a is of finite valence, at distance at least 5δ from both x and x , and a ∈ [x, x ] for some geodesic, then µ x (a)∆µ x (a) = 2.
(1) We proceed by contradiction. Assume that v is in the support of both µ x (a) and µ x (a).
In the first case, assume that no step in the flow was ending.
Then v is at distance 5δ from a and is in 4δ-geodesics between both a, x and a, x (by Proposition 1.13(2)). This is not possible.
Assume that one step of the flow for δ x was ending. If the there was no step of the flow of type ending for δ x , then the supports of µ x (a) and µ x (a) are on different spheres around a, so they are disjoint. If there was an ending step for δ x , then, since a is of finite valence, µ x (a) and µ x (a) are dirac masses carried by vertices y, y on which, respectively, y (a, x) and y (a, x ) are larger than 20δ. Since a is on a geodesic from x to x , by Proposition 1.2 one has y = y .
(2) The assumption indicates that the flow from δ x to a and from δ x to a has an ending step. It follows from Proposition 2.4 that the measures µ x (a) and µ x (a) have supports contained in two cones of parameter 160δ centered respectively on an edge starting a geodesic [a, x] and starting a geodesic [a, x ]. Let e, e these edges. One has a (e, e ) > 9(160δ) 2 . It follows from Lemma 1.8 that the intersection of the cones is reduced to {a}. Thus the intersection of the supports is empty.
Corollary 2.13. Let X be the coned-off graph of a finitely generated relatively hyperbolic group G over its family of peripheral subgroups. There exists p > 1, and ε > 0 such that for all x a∈X µ 1 (a)∆µ x (a) p is convergent, and if x is at distance at least 10δ from 1, the sum is larger than εd(1, x).
Proof. We identify G with the image of its Cayley graph in X. For the first statement, let Θ(x, x ) denote the minimal sum of angles at infinite valence vertices on a geodesic [x, x ]. Define d : X × X → N as d (g 1 , g 2 ) = d X (g 1 , g 2 ) + Θ(x, x ). By Proposition 1.6, d is coarsely larger than a word metric d w on G. In particular, there is A ≥ 1 and γ G > 1 such that S d (R, G) = {g ∈ G, d (1, g) = R} has cardinality less than Aγ R G . Denote by S d (R) the union of S d (R, G) and of the vertices x of X of infinite valence such thath d (1, x) = R. One can check that its cardinality is at most A γ R G for a certain constant A (related to the maximal order of intersection of two peripheral subgroups).
Recall that we defined a constant κ in Proposition 2.11. We choose p such that κ p < 1/γ G . Then, fixing x, we have the bound This defines a summable family of numbers therefore a∈X µ 1 (a)∆µ x (a) p is convergent.
The lower bound is given by Proposition 2.12 and the fact that there are at least d(x, x )/2 − 1 vertices of finite valence between x and x .
3. An action that is proper in the coned-off distance In this section X is the coned-off Cayley graph of a relatively hyperbolic group G, and V = RX (0) is the vector space of all functions from X (0) to R with finite support, that we endow with the 1 -norm. For all p > 1 or p = ∞, we consider W p = p (X (0) , V ). In other words, an element ω of W p is the data, for all a ∈ X (0) , of a vector ω a ∈ V , such that its norm We denote by # » IsomW the unitary group of W , i.e the subgroup of linear automorphisms of W that are isometries for the norm. The group G admits an isometric linear action on W by precomposition by isometries of X. Let us denote π : G → # » IsomW this action. We want to promote it into an action by affine isometries that has no fixed point. For that we need a cocycle, as we recall in the next subsection.
3.1. Generalities on cocycles and actions. Recall that we may see W as an affine space, and also as an abelian group (for the addition of vectors). The group of affine isometries of (the affine space) W is the semidirect product Aff IsomW = W # » IsomW for the natural action of # » IsomW on W . Given a group G, any homomorphism φ : G → Aff IsomW has an image in φ : G → # » IsomW through the quotient map, and, given this homomorphism φ, the homomorphism φ is characterised by a map c : G → W recording c(g) = φ(h)( 0 W ). Applying the law of the semidirect product reveals that c satisfy the cocycle relation c(g 1 g 2 ) = c(g 1 ) + φ(g 1 )(c(g 2 )), for all g 1 , g 2 ∈ G. The cocycle is actually the difference between φ and a given section of φ in the semi-direct product. Conversely, whenever one has φ : G → # » IsomW and c : G → W satisfying the cocycle relation for φ, one can define the map φ : G → Aff IsomW , g → φ(g) = (c(g), φ(g)), which is a homomorphism.
Definition 3.1. For a finitely generated group G, endowed with a (locally finite) word metric d, the homomorphism φ is called proper if c is proper, or equivalently if c(g i ) goes to infinity for any sequence (g i ) of elements in G for which d(1, g i ) goes to infinity in R.
If d is a left invariant metric obtained from a coned-off cayley graph over the cosets of subgroups, (or, in other words, a relative word metric), then φ is relatively proper if c(g i ) goes to infinity for any sequence (g i ) of elements in G for which d(1, g i ) goes to infinity.

3.2.
A relatively proper affine isometric action. We return to our initial context. We have G a relatively hyperbolic group, and π : G → # » IsomW for our relatively hyperbolic group G, which we want to promote to a homomorphism in AffIsomW . We thus need a cocycle for π. We identify G with the vertices of its Cayley graph Cay G in the coned-off Cayley graph X. We define the map c as follows where µ x (a) is the probability measure from Definition 2.3 and µ x (a)∆µ x (a) is the symmetric difference between the two measures µ x (a) and µ x (a) so has norm bounded above by 2, trivially. The following proves the first part of our main result, Theorem 0.1 Theorem 3.2. Let G be a relatively hyperbolic group and let X be the coned-off Cayley graph of G with respect to its peripheral subgroups. For p > 1 large enough, c has its values in W p = p (X (0) , V ), and is a cocycle for the representation π on W p .
Moreover, the homomorphism (c, π) : G → AffIsomW p is relatively proper for the metric on G induced by the coned-off Cayley graph. Precisely, there is > 0 and a constant k 0 , for which for all g ∈ G, Proof. That c is a cocycle for π is a standard computation using Proposition 2.5. The estimate on g( 0) is a consequence of Corollary 2.13.

Random coset representatives and subgroup properness
We will now recall some basic notions on induced representations from a subgroup H to a larger group G, and discuss random coset representatives (see Definition 4.3). Our main result is that in case those random coset representatives that are p -almost G-invariant exist (Definition 4.6), we can define a cocycle on G for the induced representation from H, that will have the same H-properness than the cocycle we started with (Proposition 4.14). We start by recalling some basic notions on Banach spaces.
Definition 4.1. A Banach space B is called uniformly convex if for any 0 < ≤ 2, there is δ( ) > 0 such that for any x, y ∈ B with x = y = 1 and x − y ≥ , then Suppose that a finite number of Banach spaces, B 1 , B 2 , . . . , B k are given, and that B is their product. We shall call B a uniformly convex product of the B i if the norm of an element x = (x 1 , x 2 , . . . , x k ) of B is defined by where N is a continuous non-negative function, homogeneous, strictly convex and strictly increasing.
A classical example of uniformly convex product is an p -combination. According to Clarkson [C], p -spaces are uniformly convex, and any uniformly convex product of uniformly convex Banach spaces is again uniformly convex. According to Day [D], if B is a uniformly convex Banach space, then the B-valued functions that are p are uniformly convex as well.

Induced representations.
Assume that H is a subgroup of a group G. Assume that we have a representation π H : H → # » IsomV of H in the linear isometry group of a normed vector space V in other words, a unitary representation on V ). Then, there is a normed vector space V , with an isometric embedding of V in it, and a unitary representation π G : G → # » IsomV , called the induced representation of H to G, such that the restriction to H preserves V and induces π H . We refer the reader to [BdlHV,§E.1] for the material recalled below. First, let us recall the notion of induced representation.
Definition 4.2. Given a representation (π H , V ) of a group H that happens to be a subgroup of a bigger group G, we define We can endow the vector space A ∞ with the induced action from π H π G (g) : If p : G → G/H denotes the quotient map, we further define A 0 = {ϕ ∈ A ∞ and |p(suppϕ)| < ∞} and check that the induced representation π G stabilizes A 0 . Given any norm on V , any element ϕ ∈ A ∞ induces a well-defined map This map is indeed well-defined by definition of A ∞ , and because π is a representation in the isometry group. If the norm on V is an p norm, then this map induces a norm on A 0 , by the formula that is an p -norm. One then defines A p to be the p completion of A 0 with respect to N .
Notice that for all ϕ ∈ A ∞ , for all g ∈ G, the value of ϕ at g determines the restriction of ϕ on gH. Thus any system of representatives for G/H gives an identification of A 0 with finitely V -valued supported functions on G/H, and A p is the inverse image of p (G/H, V ) under this identification. If the space V we started with is uniformly convex, then so will A p be according to the above cited result of Day [D]. When G is a relatively hyperbolic group, and H is a peripheral subgroup that acts properly on an p -space, we will use this induced representation to produce a action of G on an p -space that is proper in a sense of angles. To do that, we will need to produce a cocycle of G for this induced representation.
4.2. The example of a free product. In this subsection we are given a group H, a unitary representation (π H , V ) of H on a space V , and a cocycle c : H → V . We consider G = K * H for an arbitrary group K. We want to produce a cocycle C for G in the space A 0 for the induced representation (hence an affine isometric action of G on A 0 ).

Figure 1. Cosets seen from 1 and from γ
We consider G/H ⊆ G, the coset representatives of H in G given by normal forms in the free product. More precisely, any element g ∈ G is uniquely of the form g = k 1 h 1 . . . k n h n with k i ∈ K, h i ∈ H and k 2 , . . . , k n , h 1 , . . . h n−1 = e. For a coset gH, we write g = k 1 h 1 . . . k n h n , and we choose the representativẽ g = k 1 h 1 . . . k n . Note that if g ∈ gH, theng =g, as it should be. One can picture thatg corresponds to the projection of 1 on the coset gH for any word metric adapted to the free product. We define, for each γ ∈ G the vector C γ ∈ A 0 as follows Geometrically, for any fixed g, this records the difference (seen through the cocycle c) of the projection of 1 and the projection of γ, on the coset gH. Since we are in a free product, only finitely many such differences are non-zero, and hence the map {γ → C γ } takes its values in A 0 . We check that this is a cocycle for the induced representation.
Notice that the subspace A H of A 0 consisting of functions ϕ whose support is contained in H, is isomorphic with V , and on this subspace π G and C coincide with π H and c. We thus see that one can extend any affine isometric action of H on an p -space in an action of K * H on an p -space, such that the restrictions to each conjugate of H are all different and all isomorphic to the initial action of H.

Random representatives.
When G is a relatively hyperbolic group, and H is a peripheral subgroup, we don't have in general sufficiently stable coset representatives for H to argue like in the free product case. However, as we will see, the flow of the previous section allows to define, for each coset, a probability measure whose support is in the coset, which we think of as a random representative of the coset, that enjoy an p version of the stability of the canonical normal forms of free products. We develop here the needed vocabulary on random representatives. Definition 4.3. Given a discrete group G and a subgroup H < G, a random set of representatives for G/H is a section gH → ν gH for the map p : Proba(G) → P(G/H) that assigns to a probability measure on G, the projection of its support in G/H. We will say that random set of representatives ν is finite if all the measures ν gH have finite support, and uniformly finite if all those supports have cardinality uniformly bounded.
Notice that a set of representatives in the usual sense is a section whose image consists of Dirac masses and that for any coset gH ∈ G/H, the probability measure ν gH is supported on the coset gH ⊆ G.
Definition 4.4. For a group G and a subgroup H, a random set of representatives ν will be almost G-invariant if for any γ ∈ G, then ν gH = γν γ −1 gH except for finitely many cosets gH ∈ G/H at most.

Remarks 4.5.
(1) For a group G and a subgroup H, a random set of representatives ν will in general not be G-invariant. Indeed, G-invariance means that for any γ ∈ G and any coset gH ∈ G/H, any x ∈ gH one has ν gH (x) = ν γ −1 gH (γ −1 x) so in particular for γ ∈ H and the coset gH = H, for any x ∈ H we have ν H (x) = ν H (γ −1 x), meaning that the measure ν H is an H-invariant probability measure on H, forcing H to be finite and ν H to be the uniform measure on H.
(2) If G = K * H is a free product, then the canonical set of representatives of H-cosets, given by the normal form (and used in 4.2) gives a random set of representatives ν that assigns to each coset, the Dirac mass of the canonical representative. This is, as it should be, an almost G-invariant random set of representatives. Indeed, for any γ ∈ G and any coset gH, if we assume that g is the canonical representative, then γ −1 g is the canonical representative of γ −1 gH unless there are cancellations on the normal forms leaving an element of H in the end position. Those cosets gH are vertices on the geodesic in the Bass-Serre tree between e (the base edge of the tree) and γe, so there are only finitely many of those. (3) If G is a group obtained by a small cancellation C (λ) quotient over a free product K * H (for λ << 1), consider a generating set consisting of generators of H and of K, and the family (R i ) of relators satisfying the small cancellation condition. For each left coset gH of H we consider a geodesic from 1 to gH, and denote by g 0 ∈ gH its end point. We distinguish two cases. If the geodesic has a final subsegment labelled by a word σ appearing in a relation R i or R −1 i , of lenght at least 2λ × |R i |, then write, up to cyclic permutation R i = σhR i (or R −1 i = σhR i ) where h ∈ H and the first letter of R i is in K, and we delare that the random representative of gH consists of two dirac masses of weight 1 2 on the elements g 0 which is the end point of the geodesic, and g 0 h. If there is no such segment σ on our chosen geodesic from 1 to gH, then the random representative of gH is g 0 with probability 1. One can prove, using standard small cancellation and hyperbolicity argument (like in [RS, Appendix]), that in the later case, all geodesics from 1 to gH (and more generally all geodesics from any point to gH, provided they fellow travel [1, gH] for a sufficient length) must enter gH at g 0 . In the former case, by similar arguments, these geodesics must enter gH either on g 0 or on g 0 h.
Definition 4.6. For a group G and a subgroup H, a random set of representatives ν will be p -almost G-invariant if for any γ ∈ G, the map gH → ν gH − γν γ −1 gH 1 belongs to p (G/H).

4.4.
Cocycle from random representatives. In this subsection we will see how, given a random set of representatives for a subgroup H in a larger group G, we can construct a cocycle for the induced representation, starting from a cocycle on H. In case where the random set of representatives is p -almost G-invariant, we will see that the induced cocycle is actually on the p induced representation.
Lemma 4.7. Let G be a group and let H be a subgroup of G, with a unitary representation (π H , V ) of H and c : H → V a cocycle for (π H , V ). Then for any finite random set of representatives ν, the map defined on G by C : γ → C γ through the formula is a cocyle on A ∞ for the induced representation.
Proof. Since we assumed the random set of representatives to be finite, C γ (g) is well-defined by a finite sum. We first check that C γ (gh) = π(h −1 )C γ (g). This is a straightforward computation using first that gH = ghH, then the cocycle condition on c. Indeed since gH = ghH and γ −1 gH = γ −1 ghH, one checks Now let us check the cocycle condition, namely that C γ1γ2 = C γ1 + π * (γ 1 )C γ2 . We start, for g ∈ G, by adding and substracting z∈gH ν γ −1 Remark 4.8. If the random set of representatives ν is uniformly finite and almost G-invariant, then the cocycle defined in Lemma 4.7 is a cocycle on A 0 and on A p for any p ≥ 1. Indeed, let us check that for any γ ∈ G then C γ ∈ A 0 . Namely, we have to check that the map {gH → C γ (g) } has finite support for any fixed γ. Except for the finitely many cosets where ν gH = γν γ −1 gH , we have that for y ∈ gH ν γ −1 gH (γ −1 y)c(g −1 y) = ν gH (y)c(g −1 y) so we get that C γ (g) = 0. It is in general too optimistic to ask for almost Ginvariance, this happens for instance in the case where G = H * K or G is hyperbolic relative to H and has a small cancellation property.
Lemma 4.9. Let G be a group and let H be a subgroup of G, with a unitary representation (π, V ) of H and c : H → V a cocycle for (π, V ). Then for any uniformly finite p -almost G-invariant random set of representatives ν, the map defined on G by C : γ → C γ through the formula is a cocyle on A p for the induced representation.
Proof. We already know from Lemma 4.7 that it is a cocycle on A ∞ , so it remains only to check that for any γ ∈ G then C γ ∈ A p . Namely, we have to check that the map {gH → C γ (g) } is p-summable for any fixed γ. Because ν is p -almost Ginvariant, there are only finitely many cosets where supp(ν gH ) ∩ supp(γν γ −1 gH ) = ∅. For the other cosets, we decompose ν gH = η + ν 0 1 and ν γ −1 gH = η + ν 0 γ for η = M{ν gH , γν γ −1 gH }, so that Since C γ ∈ A ∞ , without loss of generality we can assume that g ∈ supp(ν gH ) ∩ supp(γν γ −1 gH ) = ∅, so that where R is the uniform bound on the supports of ν and hence M does not depend on g or γ. Since both ν 0 1 and ν 0 γ are bounded by ν gH − γν γ −1 gH 1 , we deduce that C γ (g) ≤ 2M ν gH − γν γ −1 gH 1 , which allows us to conclude.

Contribution and H-properness.
We define now the contribution of a coset of H in the value at γ of the cocycle C. Given two coset representatives, x for gH and x for γ −1 gH, the contribution (for these coset representatives) of gH in C γ will be the word distance in H between g −1 x and g −1 γx . It may help to think of these x and γx as projections on gH of 1 and of γ. In the context of radom coset representatives, we will sum over the probability measures.
Observe that in each term of the maximum, the only x and y that can be of interest, are those respectively in the support of ν gH and in the γ-translate of the support of ν γ −1 gH . We say that a coset gH is involved in the contribution if it realises the maximum in the formula.
Remark 4.11. Since we will be interested in whether or not the contribution tends to infinity (for a sequence of elements), the choice of d H among possible word metrics in H is often irrelevant, and we simply talk about the contribution without precising d H .
Remark 4.12. If our random set of representatives consists of Dirac masses atg (for the coset gH), as it is the case for the free products, then the (d H )-contribution is max In the case of a free product, this is the maximum of the word lengths of the elements h i of the normal form of γ.
Definition 4.13. We say that the cocycle C is H-proper if C γ goes to infinity with the (d H )-contribution of γ.
Using the geometric idea of projection, it means that if there is a coset of H on which the projection of γ is far from the projection of 1, then C γ is a large vector. We can now conclude with the main result for this section Proposition 4.14. Let G be a group and let H be a subgroup of G, and a unitary representation (π, V ) of H and c : H → V a proper cocycle for (π, V ). Then for any finite p -almost G-invariant random set of representatives ν whose supports are uniformly bounded, the A p cocycle for the induced representation defined in Lemma 4.9 is H-proper.
Proof. Let D be a bound on the diameter of the supports of ν gH . Let D c ≥ 1 be a bound on c(h) for h ∈ H such that d H (1, h) ≤ 2D.
By properness of c, we can choose R > 2D such that c(h) > 5D c for each h outside the ball of radius R in (H, d H ).
Let g such that gH is involved in the contribution for γ. We want to bound from below the norm of C γ in terms of the contribution for γ. Thus, we may assume that the diameter of the union of the supports of ν gH and γν γ −1 gH is larger than R (otherwise this contribution is less than R). Take x 0 , y 0 in those supports at maximal distance from each other (hence larger than R). The contribution of gH for γ is at least d H (g −1 x 0 , g −1 y 0 ) − 2D = d H (1, y −1 0 x 0 ) − 2D, and at most d H (1, y −1 0 x 0 ) + 2D. Computing C γ (g) we get Hence we can rewrite The first term π(g −1 y 0 )c(y −1 0 x 0 ) has norm at least 5D c by assumption on R, and the result of each sum has norm at most D c by triangular inequality (recall that the sum of the coefficients is 1) and assumption on D c . Therefore, in this case, Recall that the contribution for γ is between d H (1, y −1 0 x 0 ) − 2D and d H (1, y −1 0 x 0 ) + 2D. By properness of c the quantity c(y −1 0 x 0 ) goes to infinity with the contribution for γ, and therefore so does C γ (g) . 4.5. Peripherally proper actions. The flow of the previous section provides, in relatively hyperbolic groups, random coset representatives for the peripheral subgroups.
Proposition 4.15. Let G be a relatively hyperbolic group and let H 1 , . . . , H k be the peripheral subgroups, and let X be the coned-off Cayley graph of G with respect to H 1 , . . . , H k . Then, for each i = 1, . . . , n the map is an p -almost G-invariant random set of representatives for H i , where µ 1 ( gH i ) is the mask of the infinite valence vertex gH i from the identity 1, as in Definition 2.3. If moreover H i acts properly by affine isometries on an p -space, then G acts by affine isometries on an p -space, and this action is H i -proper.
Proof. That ν gHi = µ 1 ( gH i ) is a probability measure supported on the coset gH i is the first assertion of Proposition 2.4, hence ν is a random set of representatives for H i . That this set is p -almost G-invariant is the content of Corollary 2.13. Indeed, µ x ( gH i ) = xµ 1 ( x −1 gH i ) = xν x −1 gHi by Proposition 2.5 (since the flow step is equivariant), hence µ 1 ( gH i )∆µ x ( gH i ) = ν gH − xν x −1 gH 1 . By Corollary 2.13, this is p -summable over the set of left cosets of H. Now, we can apply Proposition 4.14, and obtain an action of G by affine isometries on an p -space that is H i -proper. Since angles at a vertex gH i are bounded above by the word-length in H i , we obtain the stated condition from the definition of H i -properness.

Proper actions of relatively hyperbolic groups
To finish the proof the our main result, Theorem 0.1, we start by a well-known remark.
Remark 5.1. If a group acts properly by affine isometries on an p -space, say p (X) for X a discrete set, then for all q ≥ p, it also acts properly on q (X). Indeed, Banach-Lamperti's theorem says that for p = 2 the linear part comes from an action of G on X, so yields a linear representation on q (X) for any q ≥ 1. Then we have that p (X) ⊆ q (X) and hence the cocycle on p can be used as is on q . Indeed, for any finite sequence (a i ) then (the inequality being Minkowski inequality for s = q/p ≥ 1) and hence on a discrete set X we have that q ≤ p , so that p (X) ⊆ q (X). Proof of Theorem 0.1. The first part of the theorem is the content of Theorem 3.2. Let H 1 , . . . , H k be the peripheral subgroups of G and assume that there are uniformly convex Banach spaces B i on which H i acts properly by affine isometries. We use, for each i = 1, . . . , n, the action obtained in Proposition 4.15 and denote by V i the uniformly convex Banach space obtained. We also use Theorem 3.2, and denote by V 0 the space obtained. Let W = k i=0 V i be a uniformly convex product of the uniformly convex Banach spaces V i , for i = 0, . . . , k. The action of G, coordinates by coordinates, is by affine isometries and it remains to check that it is proper. This amounts to checking that the action is proper in one of the coordinates at least. Let g n be a sequence in G going to infinity. According to Proposition 1.6, then either g n goes to infinity in the coned-off graph as well, or it remains bounded, and [1, g n ] gets arbitrary large angles at vertices of infinite valence. In the first case g n ( 0) goes to infinity in the V 0 coordinate. In the second case, [1, g n ] has an arbitrarily large angle at some vertex fixed by conjugates of H i(n) . Extracting so that i(n) is constant (which can be done in a way that partitions the sequence) by Proposition 4.15 g n ( 0) goes to infinity in the corresponding coordinate V i . In all cases, g n ( 0) goes to infinity in W .
If the peripheral subgroups act properly by affine isometries on some p , taking the maximum over all p's on which the peripheral subgroups act, and the p obtained in Theorem 3.2, we obtain p large enough and (according to Remark 5.1) a proper action on the p -combination of those p -spaces.
The following statement has an unfortunate technical assumption of compatibility between the CAT(0) cube complex and the system of random coset representatives, through the notion of contribution (see Section 4.4.1). The proof is completely similar and in some sense a particular case of the proof of Theorem 0.1.
Proposition 5.2. Let G be a group acting on a CAT(0) cubical complex X, with finitely many orbits, and finite edge stabilizers. Assume that the stabilizer of any point has the Haagerup property (respectively, acts properly on and p -space) and admits a random system of coset representatives that is 2 -almost G-invariant. Assume also that for any sequence (g n ) of elements of G going to infinity in G, either for x ∈ X, g n x, or the contribution for g n of cosets of stabilizers of vertices of X goes to infinity. Then G has the Haagerup property as well (respectively, G acts properly on an p -space).
Proof. If we assume the existence of 2 -almost G-invariant random coset representatives we obtain, according to Proposition 4.15 for each i = 1, . . . , k, an action on a Hilbert space (respectively, an p -space) denoted V i . We also use Niblo-Reeves' construction [NR] to obtain a Hilbert space, denoted by V 0 , from the action on a CAT(0) cube complex (and hence also on an p -space according to Remark 5.1). Let W = k i=0 V i be the 2 sum of the Hilbert spaces V i , for i = 0, . . . , k, this is again a Hilbert space (respectively, the p -sum of the p -spaces V i , which is again an p -space). The action of G, coordinates by coordinates, is by affine isometries and it remains to check that it is proper. This amounts to checking that the action is proper in one of the coordinates at least. Let g n be a sequence in G going to infinity. Assume first that there is a point x in the CAT(0) cube complex (denoted by X) so that d(x, g n (x)) goes to infinity. Then g n ( 0) goes to infinity in the V 0 coordinate. Assume now the other case, that d(x, g n (x)) remains bounded for all points x. Then by assumption, the contribution of some coset of vertex stabilizer goes to infinity.
Proof of Corollary 0.2. According to Remarks 4.5(3), in finitely presented small cancellation groups over a free product, the images of the factor groups admit an almost G-invariant random system of coset representatives. Recall also (see [MS]) that any small cancellation group over a free product acts on a cubical CAT(0) complex X with finitely many orbit, each of which has trivial stabilizer, and whose stabilizer of vertices are images of the free factors (and their conjugates). It is obtained, as in [W], by means of a wallspace structure (or hypergraph system) of an associated 2-dimensional polygonal complex. G is hyperbolic relative to these stabilizers of vertices (see for instance [P]), and by [Bow,Prop 4.13], the 1-squeleton of X is equivariantly quasi-isometric to a coned-off Cayley graph for G over these subgroups. If, for a sequence g n , and a point x ∈ X, the sequence g n x is bounded, then the sequence g n is bounded in the coned-off Cayley graph. If it goes to infinity in the group, then the contribution for it of cosets of stabilizers of vertices goes to infinity. Proposition 5.2 hence applies.