Random groups, random graphs and eigenvalues of p-Laplacians

We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine isometric, and more generally $(2-2\epsilon)^{1/2p}$-uniformly Lipschitz) with $p$ varying in an interval increasing with the set of generators. In the same model, we establish a double inequality between the maximal $p$ for which $L^p$-fixed point properties hold and the conformal dimension of the boundary. In the Gromov density model, we prove that for every $p_0 \in [2, \infty)$ for a sufficiently large number of generators and for any density larger than 1/3, a random group satisfies the fixed point property for affine actions on $L^p$-spaces that are $(2-2\epsilon)^{1/2p}$-uniformly Lipschitz, and this for every $p\in [2,p_0]$. To accomplish these goals we find new bounds on the first eigenvalue of the p-Laplacian on random graphs, using methods adapted from Kahn and Szemeredi's approach to the 2-Laplacian. These in turn lead to fixed point properties using arguments of Bourdon and Gromov, which extend to $L^p$-spaces previous results for Kazhdan's Property (T) established by Zuk and Ballmann-Swiatkowski.


Introduction
One way to study infinite groups is through their actions on various classes of spaces. From this point of view, of particular importance are the fixed point properties, that is the properties stating that a group can act isometrically on a certain type of metric space only when the action has a global fixed point. For Hilbert spaces, this is the so called property F H of J.P. Serre, which for locally compact second countable topological groups (and continuous actions) is equivalent to Kazhdan's property (T). The research around similar properties for various types of Banach spaces, or of non-positively curved spaces, has been very lively in recent years. The relevance of fixed point properties is manifest in many important areas, from combinatorics to ergodic theory, smooth dynamics, operator algebras and the Baum-Connes conjecture.
Despite their importance, many questions related to fixed point properties remain open, even in cases such as the L p -spaces, which are in a sense the closest relatives to Hilbert spaces, among the Banach spaces. In this paper we investigate fixed point properties on L p -spaces and on spaces whose finite dimensional geometry is related to that of L p -spaces, in the following sense. The research of both authors was supported in part by the EPSRC grant "Geometric and analytic aspects of infinite groups". The research of the first author was also supported by the project ANR Blanc ANR-10-BLAN 0116, acronyme GGAA, and by Labex CEMPI. Definition 1.1. Let p ∈ (0, ∞), L ≥ 1 and m ∈ N. A Banach space is said to have an L-bi-Lipschitz L p geometry above dimension m if every m-dimensional subspace of it is contained in a subspace L-bi-Lipschitz equivalent either to an ℓ p n for some n ≥ m, or to ℓ p ∞ or to some space L p (X, µ). When the above property holds for every m, we say that the Banach space has an L-bi-Lipschitz L p geometry in finite dimension.
Particular cases of spaces with L-bi-Lipschitz L p geometry in finite dimension are given by the usual spaces L p (X, µ), or spaces L-bi-Lipschitz equivalent to a L p (X, µ). Examples of spaces with L-bi-Lipschitz L p geometry above dimension m include spaces of cotype p with all subspaces of dimension k (L − ǫ)-bi-Lipschitz equivalent, for some k ≥ m and some small ǫ > 0 [BL00, Theorem G.5].
The combinatorial construction that we use in this paper has a natural connection to fixed point properties for actions on this type of spaces, which we now formulate.
Definition 1.2. A topological group Γ has property F L p m,L if every affine isometric continuous action of Γ on a Banach space with L-bi-Lipschitz L p geometry above dimension m has a global fixed point.
We say that Γ has property F L p L if every affine isometric continuous action of Γ on a Banach space with L-bi-Lipschitz L p geometry in finite dimension has a global fixed point.
The continuity condition requires simply that the orbit map g → gv is continuous, for every vector v in the considered Banach space. Recall that for p ∈ (0, 1) the metric considered on L p (X, µ) is given by the p-power of the p-norm, otherwise the triangular inequality would not be satisfied.
In the case when Definition 1.2 is restricted to isometric actions (L = 1) and the Banach spaces are only L p -spaces, the property is also called the F L p -property. A theorem of Delorme-Guichardet [Gui72,Del77] together with a standard Functional Analysis result [WW75,Theorem 4.10] imply that for every p ∈ (1, 2] property F L p is equivalent to Kazhdan's property (T) (see also [BFGM07,Theorem 1.3]). For p = 1 the equivalence is proved in [BGM12].
For p ≥ 2 property F L p implies property (T), but the converse is not true, at least not for p large. Indeed, it follows from work of P. Pansu [Pan89] and of Cornulier, Tessera & Valette [dCTV08] that given H n H , the n-dimensional hyperbolic space over the field of quaternions, its group of isometries Γ = Sp(n, 1) does not have property F L p for p > 4n + 2 (where 4n + 2 is the conformal dimension of the boundary ∂ ∞ H n H ), while the same group is known to have property (T). Also, a result of M. Bourdon [Bou16], strengthening work of G. Yu [Yu05], implies that non-elementary hyperbolic groups Γ have fixed-point-free isometric actions on an ℓ p -space for p larger than the conformal dimension of the boundary ∂ ∞ Γ (see also Bourdon & Pajot [BP03] and Nica [Nic13]). In particular this holds for hyperbolic groups with property (T).
This shows that for large p > 2 property F L p is strictly stronger than property (T). The comparison between the two properties when p > 2 is close to 2 is unclear. It is known that every group with property (T) has property F L p for p ∈ [2, 2 + ǫ), where ǫ depends on the group [BFGM07,DK16].
Like other strong versions of property (T), the family of properties F L p separates the simple Lie groups of rank one from the simple Lie groups of rank at least 2 (and their respective lattices). Indeed, all rank one groups and their uniform lattices fail to have F L p for p large enough [Yu05], while lattices in simple Lie groups of higher rank have property F L p for all p ≥ 1 [BFGM07].
Interestingly, the other possible version of property (T) in terms of L p -spaces, requiring that "almost invariant vectors imply invariant vectors for linear isometric actions", behaves quite differently with respect to the standard property (T); namely the standard property (T) is equivalent to this L p version of it, for 1 < p < ∞ [BFGM07, Theorem A]. This shows in particular that the two definitions of property (T) (i.e. the fixed point definition and the almost invariant implies invariant definition) are no longer equivalent in the setting of L p spaces, for p large.
The importance of the properties F L p comes for instance from the fact that in various rigidity results known for groups with property (T), similar results requiring weaker conditions of smoothness hold for groups with property F L p . See for instance [Nav06], where the theorem of reduction of cocycles taking values in the group of diffeomorphisms of the circle Diff 1+τ (S 1 ) to cocycles taking values in the group of rotations is true for τ = 1 p when the group has property F L p . Thus, the problem of estimating the maximal p for which a given group has property F L p is natural and useful, and several questions can be asked related to this. To begin with, we note that for every group Γ with property (T) the set F(Γ) of positive real numbers p for which Γ has F L p is open [DK16]. Let ℘(Γ) be the supremum of the set F(Γ), possibly infinite. (a) Do there exist, for any p 0 ≥ 2, groups such that F(Γ) contains (0, p 0 ) and ℘(Γ) is finite ?
(b) Do there exist groups as above that moreover fail to have F L p for all p ≥ ℘(Γ), and eventually have proper actions on L p -spaces for p ≥ ℘(Γ) ?
Up to now, the only known examples of groups with property (T) that fail to have F L p for all p larger than some p 0 are the hyperbolic groups. In particular, for hyperbolic groups the question about the geometric significance of ℘(Γ) can be made more precise.
Most examples of hyperbolic groups with property (T) come from the theory of random groups, hence it is natural to consider the questions above in the particular setting of random groups. It is what we undertake in this paper: a study of random groups from the viewpoint of the properties F L p , both in the triangular model and in the Gromov density model. 1.1. Random groups and fixed point properties. We follow the notation of [A LŚ15]. Also, in what follows we write f ≃ g for two real functions f, g defined on a subset A ⊆ R if there exists C > 0 such that f (a) ≤ Cg(Ca + C) and g(a) ≤ Cf (Ca + C), ∀a ∈ A.
The source of the theory of random groups is in the work of Gromov [Gro93,Gro03], and in the context of the triangular model and of property (T) it has been reformulated byŻuk [Ż03].
The triangular model of random groups that appears the most often in the literature is the triangular density model M(m, d), defined for a density d ∈ (0, 1). This is the model in which, for a fixed set of generators S, with |S| = m, a set of (2m − 1) 3d relations R is chosen uniformly and independently at random, among all the subsets of this cardinality in the full set of cyclically reduced relators of length 3 (with cardinality ≃ m 3 ). (As is standard, quantities such as (2m − 1) 3d are rounded to the nearest integer.) The groups Γ = S|R are the elements composing the model. For more details on this model, we refer to [Ż03] and [KK13].
A variation of this model, which is an analog for random groups of the Erdös-Renyi model of random graphs, is the following.
Definition 1.5. Let ρ be a function defined on N and taking values in (0, 1). For every m ∈ N, the binomial triangular model Γ(m, ρ) is defined by taking a finite set of generators S with |S| = m, and groups Γ = S|R , where R is a subset of the set of all ≃ m 3 possible cyclically reduced relators of length 3, each relator chosen independently with probability ρ(m).
One of our main theorems is the following.
Theorem 1.6. For any δ > 0 there exists C > 0 so that for ρ = ρ(m) ≥ m δ /m 2 , and for every ε > 0 a.a.s. a random group in the binomial triangular model Γ(m, ρ) has F L p (2−2ε) 1/2p for every p ∈ 2, C(log m/ log log m) 1/2 . In particular, a.a.s. we have F L p for all p in this range.
The model Γ(m, ρ) is closely related to the density model M(m, d), when ρm 3 ≃ (2m − 1) 3d . Property F L p is preserved by quotients, in particular by adding more relations, so it is a "monotone property" in the sense of [J LR00, Proposition 1.13] (see Section 10). Thus, general results on random structures mean that our theorem implies the following in the density model M(m, d).
Corollary 1.7. For any fixed density d > 1/3 there exists C > 0 so that for every ε > 0 a.a.s. a random group in the triangular density model M(m, d) has F L p (2−2ε) 1/2p for every p ∈ 2, C(log m/ log log m) 1/2 . In particular, a.a.s. we have F L p for all p in this range.
In the case of F L 2 , that is, property (T), this is a result ofŻuk [Ż03], with steps clarified by Kotowski-Kotowski [KK13].
Note that for any density d < 1/2 a random group in M(m, d) is hyperbolic [Ż03].
The picture drawn by Theorem 1.6 is completed by the results of Antoniuk, Luczak andŚwiatkowski [A LŚ15], improving previous estimates ofŻuk [Ż03], and stating that: • there exists a constant κ such that if ρ ≤ κ m 2 , then a.a.s. a group in the model Γ(m, ρ) is free; , then a.a.s. a group in Γ(m, ρ) is neither free nor with property (T); • there exists a constant κ 3 such that if ρ ≥ κ3 log m m 2 , then a.a.s. a group in Γ(m, ρ) has property (T).
In the first two of these cases, the failure of property (T) implies that the groups have none of the F L p properties [BFGM07]. In the third case we show a result like Theorem 1.6, with a bound growing a little slower than (log log m) 1/2 , see Theorem 9.4.
As far as Corollary 1.7 is concerned,Żuk had already proven [Ż03] that for any density d < 1/3, a random group in the triangular density model M(m, d) had free factors, and hence property (T) and all F L p properties fail. We give some partial information at d = 1/3 in Section 10. Note that the results in [A LŚ15] for the first two cases do not immediately apply here, since the properties they deal with are not monotone.
Another model for random groups is the Gromov density model G(k, l, d), where for a fixed set of generators A with |A| = k, a random group in the model is a group Γ = A|R with presentation defined by a collection R of reduced relators of length l, R of cardinality (2k − 1) dl , chosen randomly with uniform probability.
Like for the triangular model, there exists a version of this that is closer to Erdös-Renyi for graphs.
Definition 1.8 (Gromov binomial model). Fix a number of generators k ≥ 2, a set of generators A with |A| = k, and a function ρ : N → (0, 1).
A group Γ = A|R in the k-generated Gromov binomial model G(k, l, ρ) is defined by taking R a collection of cyclically reduced relators of length l in the alphabet A, each chosen independently with probability ρ(l).
Remark 1.9. For a fixed number of generators k ≥ 2, a fixed density d ∈ (0, 1), and the function ρ(l) = (2k − 1) −(1−d)l , the model G(k, l, ρ) is closely related to the Gromov density model G(k, l, d), since there are ≍ (2k − 1) dl cyclically reduced words of length l in R, where A ≍ B means 1 C A ≤ B ≤ CA, for some constant C > 0. See Section 10 for more details.
In the density model G(k, l, d) as well, when d < 1/2 a random group is nonelementary hyperbolic [Gro93, Chapter 9].
We prove the following.
Theorem 1.10. Choose p ≥ 2, ǫ > 0 arbitrary small and k ≥ 10 · 2 p . Fix a density d > 1/3. Then a.a.s. a random group in the Gromov density model G(k, l, d) has In particular, a.a.s. we have F L p ′ for all p ′ in this range.
This follows from the corresponding theorem in the Gromov binomial model G(k, l, ρ), see Theorem 12.1. For any fixed k ≥ 2 is it natural to expect a result where p → ∞ as in Theorem 1.6, however our methods currently do not show this. We do find a new proof of property (T) for any fixed k ≥ 2 and d > 1/3, which moreover applies at d = 1/3 as well (see Theorem 12.6 for a precise statement).
Previous progress on the problem of F L p -properties with p > 2 for random groups in the Gromov density model had been made by P. Nowak in [Now15] (see Remark 1.12).
In the class of groups with property (T), the subclass of hyperbolic groups plays a special role, since by [Oll05, §III.3] and [dC05] every countable group with property (T) is the quotient of a torsion-free hyperbolic group with property (T). Therefore, Theorems 1.6 and 1.10 may be seen as an indication that the generic countable groups with property (T) also have F L p for p in an arbitrarily large interval (2, p 0 ).
1.2. Conformal dimension. Another setting emphasizing the interest of the properties F L p lies in their connection with P. Pansu's conformal dimension. For a hyperbolic group Γ, the boundary ∂ ∞ Γ comes with a canonical family of metrics; the infimal Hausdorff dimension among these is the conformal dimension Confdim(∂ ∞ Γ). This is an invariant of the group, and in fact, if two hyperbolic groups are quasi-isometric then they have the same conformal dimension. For more details, see [MT10].
Conformal dimension can sometimes be used to distinguish hyperbolic groups even if their boundaries are homeomorphic, see Bourdon [Bou97]. For random groups in the Gromov density model at densities d < 1/8, the second author has found sharp asymptotics for the conformal dimension using small cancellation methods [Mac12,Mac16].
However, small cancellation methods completely fail for random groups at densities d > 1/4, and certainly do not work for random groups in the triangular models. Therefore it is of interest that we are able to bound the conformal dimension in a new way at densities d > 1/3 using the F L p properties.
As mentioned above, Bourdon showed that if a Gromov hyperbolic group has property F L p for some p > 0, then the conformal dimension of its boundary is at least p. A consequence of this inequality, an upper bound computation, and Corollary 1.7 is the following.
Remark 1.12. P. Nowak has also obtained a lower bound for the parameter ℘(Γ) and hence conformal dimension, that can be explicitly calculated, in the triangular and in the Gromov density models, using spectral methods [Now15, Corollary 6.4]. However, his bound is an explicit decreasing function slightly larger than 2.
Remark 1.13. Theorem 1.11 provides in particular a positive answer to Question 1.3(a). The first such example, also among hyperbolic groups, was provided by Naor and Silberman [NS11, Theorem 1.1]. Theorem 1.11 brings the additional information that the situation described in Question 1.3(a), is in fact generic in this standard model of random groups. In view of the remark following Theorem 1.10, it is expected that this same situation is generic for the whole class of countable groups with property (T).
Remark 1.14. A consequence of Theorem 1.11 is that for a generic hyperbolic group Γ in the model M(m, d) with d ∈ ( 1 3 , 1 2 ), there exists a constant κ = κ(d) such that where ǫ > 0 is fixed. This illustrates that a formula relating ℘(Γ) and Confdim(∂ ∞ Γ) is plausible, in particular an equality as conjectured in Question 1.4.

Random graphs and strong expansion.
Our results on random groups rely on spectral results on random graphs. Indeed, every finitely presented group Γ has a presentation in which all relators are of length three, and every such presentation yields an action of Γ on a simplicial 2-complex X, the Cayley complex. The link of every vertex is a graph L(S), and if the smallest positive eigenvalue λ 1 (L(S)) of the Laplacian of this graph satisfies λ 1 (L(S)) > 1 2 , then Γ has property (T). This has been shown byŻuk and Ballmann-Światkowski [Ż03,BS97], and appears implicitly in [Gro03]. (See Sections 2 and 8.) In the case of a random group Γ ∈ Γ(m, ρ), the link graph is nearly a union of three random graphs coming from a suitable random graph model.
There is a large literature on the first positive eigenvalue of the Laplacian of a random graph. In the case of constant degree the problem is equivalent to bounding the second largest eigenvalue of the adjacency matrix, and this opens up methods used by Friedman to give very precise asymptotics. In our context the bound λ 1 (L(S)) > 1 2 follows from a result of Friedman and Kahn-Szemerédi [FKS89] that random graphs have λ 1 close to 1.
In our setting we must replace the Laplacian by a non-linear generalization of it, the p-Laplacian, for p ∈ (1, ∞), see Section 2. The p-Laplacian has been used in combinatorics and computer science [BH09] and turns out to be a useful tool for estimates of the L p -distortion [JV13].
We use a sufficient condition for property F L p L , described in the theorem below, which can be obtained by slightly modifying arguments of Bourdon. The latter arguments use Garland's method of harmonic maps, initiated in [Gar73], developed byŻuk [Żuk96] and Wang [Wan98], and further used and developed by Ballmann Here, given a graph L we denote by λ 1,p (L) of a graph L the first positive eigenvalue of the p-Laplacian of L (Definition 2.1).
Theorem 1.16 (Bourdon [Bou12]). Suppose p ∈ (1, ∞) and X is a simplicial 2complex where the link L(x) of every vertex x has λ 1,p (L(x)) > 1 − ε and has at most m vertices. If some group Γ acts on X simplicially, properly, and cocompactly, then Γ has the property F L p m+1,(2−2ε) 1/2p . Bounding λ 1,p (L) away from zero corresponds to showing that L is an expander, but in changing p we can lose a lot of control, see Proposition 11.6. So to show that λ 1,p is as close to 1 as we wish, we have to prove new results for random graphs.
Given m ∈ N and ρ ∈ [0, 1], let G(m, ρ) be the model of simple random graphs on m vertices, where each pair of vertices is connected by an edge with probability ρ.
For every m ∈ N and every ρ satisfying κ log m m ≤ ρ ≤ χ(m)m 1/3 m we have that with probability at least 1 − C ′ m ξ a graph G ∈ G(m, ρ) satisfies The methods of Friedman are not available in this non-linear situation, but Kahn-Szemerédi's approach does adapt, as we discuss further in Section 3.
1.4. Recent result for a larger class of Banach spaces. About a year after this paper has been finished, Tim de Laat and Mikael de la Salle proved in [dLdS17] that, given a uniformly curved Banach space X, for any density d > 1 3 , a.a.s. a random group in the triangular density model M(m, d) has the fixed point property F X (i.e. every action of such a group by affine isometries on X has a global fixed point). Uniformly curved Banach spaces were introduced by G. Pisier in [Pis10], examples of such spaces are L p -spaces, interpolation spaces between a Hilbert and a Banach space, their subspaces and equivalent renormings. A uniformly curved space X has the following key property. Given a finite graph G with set of vertices G 0 and set of edges G 1 , G 0 equipped with the stationary probability measure ν for the random walk on G, defined by ν(x) = val(x) y∈G 0 val(y) (where val(x) denotes the valency of the vertex x), the norm of the Markov operator A G on L 2 0 (G 0 , ν; X) is small provided that the norm of the Markov operator A G on L 2 0 (G 0 , ν) is small. (Here by L 2 0 we mean square integrable functions with expectation zero.) The outline of the proof of the de Laat-de la Salle theorem is as follows. They use, likeŻuk [Ż03] and Kotowski-Kotowski [KK13], the permutation model for groups and, correspondingly, the configuration model for random graphs, and a theorem of Friedman stating that for a random graph G in the latter model a.a.s. the norm of the Markov operator A G on L 2 0 (G 0 , ν) is small, provided that the number of permutations taken is large enough.
It follows that, for a random group Γ in the permutation model (with a large enough number of permutations), given the simplicial complex ∆ Γ of the corresponding triangular presentation of Γ, a.a.s. for every vertex link L in ∆ Γ , the norm of the Markov operator A L on L 2 0 (L 0 , ν; X) is small, uniformly in L. The space X being uniformly curved, it is also superreflexive, hence by a result of Pisier [Pis75] it admits an equivalent norm that is p-uniformly convex, for some p ∈ [2, ∞), and preserved by the isometries of the initial norm on the space X. Thus the statement is reduced to the case when X is p-uniformly convex, and Γ is a group that acts properly discontinuously cocompactly on a simplicial complex ∆ Γ with the property that for all the vertex links L the Markov operators A L on L 2 0 (L 0 , ν; X) have uniformly small norms. An adaptation of an argument of Oppenheim [Opp14] can then be applied to conclude that the random group Γ must have property F X .
In the particular case of L p -spaces the theorem of de Laat-de la Salle gives that for every p 0 ≥ 2, for any density d ∈ 1 3 , 1 2 , a.a.s. a group Γ in the triangular model M(m, d) satisfies all the fixed point properties F L p with p ∈ (0, p 0 ]. We believe that if, instead of using the permutation model for groups, respectively, the configuration model for random graphs, and Friedman's Theorem, de Laatde la Salle would use the binomial triangular model for groups, the Erdös-Renyi model for random graphs and the estimate in Theorem 1.17 for p = 2, then they would obtain a version of Corollary 1.7 with a slightly larger interval, that is with p ∈ [2, C(log m) 1 2 ]. Therefore, in Theorem 1.11 one could suppress the log log m from the denominator in the lower bound, and in Remark 1.14 the exponent in the first term in (1.15) would become 1 2 instead of 1 2 − ǫ.
We think nevertheless that the proof provided in this paper has its own intrinsic value, firstly because it relies on elementary mathematics only, it is self contained and independent of Pisier's results, and secondly because we find it intriguing that by two different approaches approximately the same lower bound estimates are obtained. This may suggest that the first inequalities in Theorem 1.11 and Remark 1.14 may in fact be asymptotic equalities.
1.5. Plan of the paper. Section 2 is an introduction to the p-Laplacian, with several interpretations and estimates of its first non-zero eigenvalue.
In Sections 3 to 7, Theorem 1.17 is proven, by reducing the problem to a small enough upper bound to be obtained for a finite number of sums varying with the set of vertices, then by splitting each sum into light and heavy terms, and estimating separately the two sums of light, respectively heavy terms.
Section 8 links values of λ 1,p to the properties F L p m,L . This is then used in Section 9 to show the results on random groups in the triangular model, deduced from Theorem 1.17.
We describe how to use monotonicity to switch between models, and the application to conformal dimension in Section 10.
In Sections 11 and 12, the same strategy is applied to prove a similar result of generic p-expansion for multi-partite graphs, and the latter is then applied to prove Theorem 1.10.
1.6. Notation. We use the standard asymptotic notation, which we now recall. When f and g are both real-valued functions of one real variable, we write f = O(g) to mean that there exists a constant L > 0 such that f (x) ≤ Lg(x) for every x; in particular f = O(1) means that f is uniformly bounded, and f = g + O(1) means that f − g is uniformly bounded. The notation f = o(g) means that lim x→∞ f (x) g(x) = 0 . 1.7. Acknowledgements. We thank Damian Orlef for pointing out an issue with the p ∈ (2, 3) case in an earlier version.

Eigenvalues of p-Laplacians
In what follows G is a graph, possibly with loops and multiple edges (a multigraph). When the graph has no loops or multiple edges and the edges are considered without orientation we call it simple. Let G 0 be its set of vertices and G 1 its set of edges. Given two vertices u, v, we write u ∼ v if there exists (at least) one edge with endpoints u, v and we say that u, v are neighbours.
Fix an arbitrary orientation on the edges of G, so that each edge e ∈ G 1 has an initial endpoint e − and a target endpoint e + in G 0 . Given a function x : G 0 → R, the total derivative of x is defined as dx : For e ∈ G 1 , we write the unordered set (with multiplicities) of endpoints of e as V(e) = {e − , e + }. Observe that |dx(e)|, or indeed any symmetric function of e − and e + , is independent of the choice of orientation of e ∈ G 1 .
Fix p ∈ (1, ∞). Given x ∈ R, we define {x} p−1 = sign(x)|x| p−1 when x = 0, and we set {0} p−1 = 0. The graph p-Laplacian on G (see [Amg03,BH09]) is an operator from R G0 to R G0 defined by where val(u) is the valency of u. The operator ∆ p is linear only when p = 2. Still, by abuse of language, one can define eigenvalues and eigenfunctions which serve the purpose in the L p -setting as well.
Definition 2.1. We say λ ∈ R is an eigenvalue of ∆ p for G if there exists a non-zero function x ∈ R G0 so that ∆ p x = λ{x} p−1 . We call such a function x an eigenfunction of ∆ p .
We denote by λ 1,p (G) the smallest eigenvalue of ∆ p which corresponds to a nonconstant eigenfunction.
The standard (normalised) graph Laplacian ∆ = ∆ 2 can equivalently be defined using a weighted inner product on R G0 . Consider the degree sequence d = (d u ) ∈ N G0 , d u = val(u), and define x, y d = u∈G0 x u y u d u . Then for x ∈ R G0 , ∆ is the linear operator such that x, ∆x d = dx 2 2 , where the norm on the right hand side is on R G1 . When the right hand side becomes dx p p = e∈G1 |dx(e)| p , the same equality defines ∆ p [BH09, Section 3]; consequently all eigenvalues are ≥ 0. Note that in [BH09, Section 3] the equality The reason is that in that paper, what stands for dx p p , also denoted by Q p (f ), is a sum where each term |x(e + )−x(e − )| p appears twice (in other words, no orientation is chosen on the edges).
The value of λ 1,p for a multigraph G may be calculated as follows. The Poincaré p-constant π p is defined as in the classical case to be the minimal constant π such that for every function x ∈ R G0 , inf c∈R u∈G0 We will use the following Rayleigh Quotient characterisation of λ 1,p (G) [Amg03, Theorem 1], see also [BH09, Theorem 3.2] and [Bou12, Proposition 1.2]. Note that the constant functions are eigenfunctions with eigenvalues 0, and that for every p > 1, the minimal eigenvalue for non-constant functions, λ 1,p (G), is 0 if and only if G is disconnected; in this case we interpret π p as ∞.
Proposition 2.2. Fix p ∈ (1, ∞) and a multigraph G. Then 2.1. Varying p. Later we need the following estimate on how λ 1,p (G) varies as a function of p.
Lemma 2.7. For a graph G, λ 1,p (G) is a right lower semi-continuous function of p. To be precise, for p ≥ p ′ ≥ 2, where E is the number of edges in G.
Proof. Let x ∈ R G0 be a non-constant function which attains λ 1,p (G) in (2.3), i.e., Let y be a scaled copy of x ′ so that where the last equality follows from u∈G0 {y u } p ′ −1 val(u) = 0. In particular, for each u ∈ G 0 , |y u | ≤ 1 and thus |y u | p ≤ |y u | p ′ . So

Now Hölder's inequality gives
Bounding λ 1,p for random graphs Given m ∈ N and ρ ∈ [0, 1], recall that a random graph in the model G(m, ρ) is a simple graph on m vertices, with each pair of vertices connected by an edge with probability ρ.
Our goal, from now until the end of Section 7, is to show the following bound on λ 1,p for a random graph in this model. Theorem 1.17. Given a function χ : N → (0, ∞) with lim m→∞ χ(m) = 0, for every ξ > 0 and every p ≥ 2 there exists positive constants κ = κ(ξ), C = C(ξ) and C ′ = C ′ (ξ, χ), such that the following holds.
For every m ∈ N and every ρ satisfying In fact, we prove lower bounds on λ 1,p (G) when G is chosen from a more restrictive random graph model, denote a sequence of vertex degrees, where we assume that d i is even (a necessary condition). The random graph model G(m, d) is defined by letting G ∈ G(m, d) be chosen uniformly at random from all simple graphs with this degree sequence. For example, in the case that d i = d for all i, this is the model of random d-regular graphs.
Theorem 3.1. Consider a constant θ ≥ 1 and a function χ : Then for every ξ > 0 there exists C and C ′ depending on θ, ξ, with C ′ moreover depending on the function χ, so that for every m ∈ N and p ≥ 2, and every degree sequence d ∈ N m with i d i even and Theorem 3.1 implies the result in G(m, ρ), when combined with the following lemma.
To approach Theorem 3.1, we consider again the characterisation (2.5) of λ 1,p when we have a fixed degree sequence This motivates the following notation.
Notation 3.5. Given a real number p ≥ 2 and two real numbers a, b, we define ℜ p (a, b) = |a| p + |b| p − |a − b| p .
Using this notation we can write In the particular case that Therefore, to prove Theorem 3.1 it suffices to show that with high probability X x (G) is bounded from above by a suitable uniform small positive term, for all x ∈ S p,d (G 0 ). This is proved using a variation of the Kahn-Szemerédi method [FKS89] for bounding λ 1,2 , which is roughly as follows: every x ∈ S p,d (G 0 ) can be approximated by some function x ′ in a suitable finite net, and if the approximation is accurate enough then it suffices to show that with high enough probability X x ′ (G) has a uniform small positive upper bound for every x ′ in the net, see Section 4. The reason for switching from the Erdös-Renyi model G(m, ρ) to the prescribed degree model G(m, d) is that in our case this net is defined in terms of the vertex degrees d. For each point x ′ in this net, the terms in X x ′ (G) split into small and large values, and the two contributions are bounded independently. We discuss this further in sections 5-7.
We remark that Kahn and Szemerédi worked in the permutation model for random regular graphs. However, their method was adapted to the model G(m, d) by Broder-Frieze-Suen-Upfal [BFSU99, Theorem 7], and it is their proof that we follow more closely.

Approximating on finite sets
In this section we define a net of points approximating well enough the points in the set S p,d , we provide bounds on the size of this net, and we show that good enough bounds on an infimum defined as in (2.5) but with S p,d replaced by the net suffice to bound λ 1,p .
4.1. The net and its size. Suppose we have a graph G with vertex set G 0 = {1, 2, . . . , m} and degree sequence d For any R ≥ 1 and small enough constant ǫ > 0, we define a corresponding finite net that will be used to approximate S p,d (G 0 ).
Here we follow: Convention 4.1. We let q denote the Hölder conjugate p p−1 of p. Throughout all that follows, to simplify estimates we assume ǫ satisfies: Assumption 4.2. We have ǫθ ≤ 1.
Recall that θ ≥ max d i /d j and is close to 1 in our applications. Later we will take R = (1 + ǫθ 1/p ) q , which by Assumption 4.2 satisfies R ≤ 4. We need to know the size of T p,d,R (G 0 ). Before we bound this, it is helpful to recall the following.
Lemma 4.3. There exists m 0 so that for all p ≥ 2 and m ≥ m 0 , the volume V q (R) of the radius R ball in R m endowed with the norm · q is bounded by Proof. It suffices to consider R = 1, where Applying this to (4.4), we see that Proof. Consider the set by Assumption 4.2. This ball B is an affine transformation of the ball V q (R ′ ), so by Lemma 4.3 it has volume We combine (4.6) and (4.7) to conclude:

4.2.
Bounds on the net suffice. The following proposition shows that to bound We now bound the size of x ′ in the norm · p,d , using weighted Hölder's inequalities.
and so Recall that, by construction, for each i we have . Since x i and x ′ i are either both non-positive or both non-negative, we find that By the Mean Value Theorem applied to t → |t| p , for each e ∈ G 1 there exists t(e) ∈ R with |t(e)| ≤ 2(ǫθ) 1/(p−1) /(dm) 1/p so that Therefore, by Hölder's and Minkowski's inequalities, and our assumption Z x (G) ≤ 1, (4.10) The final remark follows from the following argument.
4.3. Preliminary bounds. The quantity ℜ p (a, b) = |a| p + |b| p − |a − b| p is difficult to work with, so in the following sections we have occasions to use more convenient quantities, described below.
Notation 4.11. Given a real number p ≥ 2 and two real numbers a, b, we define where x ∨ y denotes the maximum of x and y.
Proposition 4.12. For every a, b ∈ R the following hold. (4.14) Moreover, for p ≥ 3 we have Proof. For every a, b ∈ R and λ > 0, and for It therefore suffices to show all inequalities for a = 1 and −1 ≤ b ≤ 1. In this case (4.13) The second inequality is immediate: we apply the Mean Value Theorem to the function t → t p to find x between 1 and 1−b > 0 so that 1−(1−b) p = bpx p−1 ; in particular 0 ≤ x ≤ 2. Therefore, We now prove the first inequality.
so the Lagrange form of the remainder in Taylor's theorem gives that there exists y ∈ (1, 1 + c) with  In what follows we frequently drop the index p ≥ 2 from Notations 3.5 and 4.11.

Bounding X on the net
In what follows we always assume that the sequence of vertex degrees The set of edges of G splits into two subsets with respect to the function x, the light and respectively heavy edges, whose definitions depend on a parameter β = p/(2 + 2p).
Consequently X x (G) splits into two sums, of light and respectively heavy terms: We have the following bounds on light and heavy terms.
and moreover this probability holds on a set in G(m, d) defined independently of p.
We postpone the proofs of Propositions 5.1 and 5.2 until sections 6 and 7, respectively, and in the remainder of this section we use these two propositions to prove Theorem 3.1. 5.1. Proof for a single p. We begin by finding a high probability bound on λ 1,p for a single value of p = p(m).
As we are not trying to optimise for small p ≥ 2, we use 2 + 2p ≤ 3p ≤ 2p 2 , p 2 ≤ 2p 2 and p(p + 1) ≤ 2p 2 to find (5.4) To have the probability (5.3) going to one, it suffices that for then the lower bound in (5.3) is at least 1 − 2e −m+o(m) − o(m −ξ ). We choose a suitably large constant C 2 so that for K = C 2 (1 + log(d)/p) we have .
A brief calculus estimate shows this is maximised for p = 1 6 log(d), and so is bounded by a constant. Consequently, d −1/3p log(d)/p is bounded by a multiple of d −1/2p 2 .
Applying this to (5.4) we see that, for some C 3 , Y x (G) ≤ C 3 · p 3 d 1/2p 2 . and this holds for all x ∈ S p,d (G 0 ) with probability at least 1 with the same probability.

Simultaneous bounds in p.
We now get bounds on λ 1,p (G) which hold for a range of values of p simultaneously. Recall from (5.3),(5.6) above that for any particular choice of 2 ≤ p ′ ≤ p, and the choice of K as in (5.5), we have where we used ǫ ≥ d −1/p . The last term o(m −ξ ) comes via the heavy bound Proposition 5.2 from Lemma 7.2. This lemma describes properties of G ∈ G(m, d) independent of p, so our heavy bounds will hold a.a.s. for all 2 ≤ p ′ ≤ p. (The light bounds, however, are not independent of p, so the probabilities here decrease as we get bounds for more and more values of p.) Suppose we fix 2 = p 0 < p 1 < . . . < p L = p, with p i+1 /p i − 1 bounded by a constant τ ∈ (0, 1) and L ≤ log(p/2)/ log(1 + τ ) + 1. By (5.7) we have that for all i = 0, . . . , L − 1 simultaneously, with probability at least Since the number of edges in G is at most dm, Lemma 2.7 gives that with probability at least as in (5.8), we have for all i = 1, . . . , L and for all If we choose (p i ) so that some p j equals 3, then we may assume that I pi<3 = I p ′ <3 in the estimate above. We have Set τ = (log(dm)) −1 p 4 ·d −1/2p 2 , and then since we may assume that 1−τ log(dm) ≥ 0 we conclude that for all 2 ≤ p ′ ≤ p , It remains to bound the probability that this holds, using the lower bound (5.8). We can assume that τ < 1. Therefore, log(1 + τ ) ≥ τ 2 and L ≤ log(p/2)/ log(1 + τ ) + 1 ≤ 2 τ log(p/2) + 1. If L = 1 we are done, so assume that 2 τ log(p/2) ≥ 1 and so L ≤ 4 τ log(p/2). So by (5.8), our probability of failure is at most o(m −ξ ) plus 8 exp log log( 1 2 p) − log(τ ) + − 1 6000 so our probability of failure is at most By our choice of K in (5.5), this probability is ≤ 8e −m+o(m) , and so Theorem 1.17 is proved.

Bounding light terms
The aim of this section is to prove Proposition 5.1, the bound on the contribution X l x (G) of the light edges to X x (G). Rather than working directly in the model G(m, d), we use the configuration model G * (m, d) (for an overview, see [Wor99]). In this model the vertex set is is a graph with vertex set F 0 , and edge set F 1 a perfect matching of F 0 , chosen uniformly at random from all such perfect matchings.
Given Recall that X x (G), X l x (G) and X h x (G) sum ℜ(x u , x v ) over endpoints of certain edges in G. Let X x (G), X l x (G), X h x (G) be the corresponding sums where ℜ is replaced by ℜ, and X x (G), X We define X x , X l x and X h x on G * (m, d) by extending the definition from G ∈ G(m, d) to M (F ), where F ∈ G * (m, d). To be precise, define Likewise, let E h = F 1 \ E l , and define X h x (F ) analogously. For p ≥ 3, by (4.15), X l x (G) ≤ p X l x (G). To bound X l x (G), we first show that for a fixed x and for F ∈ G * (m, d) both E( X l x ) and X l x (F ) − E( X l x ) have small upper bounds uniform in x ∈ T p,d,R (G 0 ), with probability close to 1. For p ∈ [2, 3] we use a variation on this to show that both EX l x and X l x (F ) − E(X l x ) are small. The bound on the size of T p,d,R ({1, . . . , m}) given by Proposition 4.5 then implies that this same bound holds with probability close to 1 for all such x. Finally, Proposition 6.1 gives the bound for G ∈ G(m, d). Further details are provided in Subsection 6.4. 6.1. Bounding the expected value for p ≥ 3.
To show this bound in expected value, the key step is the following lemma.
Proof. Let LHS denote the left hand side of (6.4).
. Therefore, by Hölder's inequality, Clearly, the map V 1 → V defined by (i, j) → i is at most m to 1. On the other hand, the map V 1 → V defined by (i, j) → j is at most Aθm/γ to 1, since γ/dm ≤ |x i | p−1 |x j | ≤ |x i | p implies that there are at most Aθm/γ possible values for i because So, using Lemma 6.3, we have Similarly to above, by (4.13) (using 1 + p2 p−1 ≤ 13) and Lemma 6.3 we have: We use the lower bound (2.4) for dx p for the complete random graph K m , given the value λ 1,p (K m ) = m−2+2 p−1 m−1 found by Amghibech (Theorem 11.1).
Combining our bounds, we have 6.3. Light terms close to expected value. Our next goal is to prove that, for fixed x ∈ T p,d,R , X l x is very close to its expected value. Proposition 6.6. For any α ∈ (0, 1), so that 2β +2α ≤ 1, and any positive number K > 0, the following inequality holds for every x ∈ T p,d,R (M (F ) 0 ), The proof of this fact is similar in spirit to [FKS89], but we prove a weaker statement than they do, which suffices for our purposes.
We order the vertices of F 0 lexicographically: (i, s) < (j, t) if i < j, or if i = j and s < t. We now define a martingale on G * (m, d) by exposing the edges of F sequentially. First we reveal the edge connected to (1, 1), then the edge connected to the lowest remaining unconnected vertex, and so on. This defines a filter (F k ), where F k is the σ-algebra generated by the first k exposed edges.
Let S k = E( X l x |F k ). Then S 0 = E( X l x ), and at the end of the process we have S E = X l x . To apply standard concentration estimates to S E − S 0 , we need to have control on the size of S k − S k−1 .
For simplicity, given e ∈ F 1 with V(e) = {a, b}, we write Thus X l x (F ) = e∈F1 ℜ l (e). For F, F ′ ∈ G * (m, d), we write F ≡ k F ′ if and only if F and F ′ lie in the same subsets of F k , i.e., F and F ′ have the same first k edges.
For a given F ∈ G * (m, d), we bound |S k (F ) − S k−1 (F )| using a switching argument (compare Wormald [Wor99, Section 2]). Suppose the kth edge of F joins a 1 to a 2 . Let J ⊂ F 0 be {a 2 } union the set of endpoints of the remaining E − k edges. For each b ∈ J, let S b be the collection of F ′ ∈ G * (m, d) so that F ′ ≡ k−1 F and F ′ joins a 1 to b. Then For each b ∈ J, there is a bijection between S a2 and S b defined as follows: for F ′ ∈ S a2 which joins a 3 to b, define F ′′ ∈ S b by deleting {a 1 , a 2 }, {a 3 , b} from F ′ and adding {a 1 , b}, {a 3 , a 2 }. Since only at most two values of ℜ l (e) change, and | ℜ l (e)| ≤ 2d β /dm for any edge e, we have | X l With this, we can apply Azuma's inequality.
Theorem 6.7 ([J LR00, Theorem 2.25]). If (S k ) N k=0 is a martingale with S n = X and S 0 = EX, and there exists c > 0 so that for each 0 < k ≤ N , |S k − S k−1 | ≤ c, then Proof of Proposition 6.6. We apply Theorem 6.7 to (S k ) with N = E, T = K/d α , c = 8d β /dm, to get where we use that E ≤ dm and 2α + 2β ≤ 1.
Applying this to (6.9) proves Proposition 5.1 for p ≥ 3 since in that case X l x ≤ pX l x , d 2 = o(m 2/3 ), and R ≤ 4. The p ∈ [2, 3] case follows a similar argument, using the bounds of Lemma 6.5 and Proposition 6.8.

Bounding heavy terms
In this section our goal is to prove, adjusting notation slightly, the following bound on the heavy terms of X x (G).
Proposition 5.2. For β = p/(2 + 2p) and d = o(m 1/2 ) we have that for any ξ > 0, there exists C ′ = C ′ (θ, ξ) so that and moreover this probability holds on a set in G(m, d) defined independently of p.
We use (4.14) to see that x v ) > d β /dm}, and β = p/(2 + 2p). We will bound X h x by showing that if we can control the number of edges between subsets of a graph, then X h x has an explicit bound. As previously, in what follows θ ≥ 1 is a fixed constant.
Definition 7.1. Let G be a graph with |G 0 | = m vertices, minimum degree d min and maximum degree d = d max such that θ ≥ d/d min .
Given subsets of vertices A, B ⊂ G 0 , denote by E A,B (G) the number of edges in G between A and B, and set µ(A, B) = θ|A||B|d/m.
We say that G has (θ, C)-controlled edge density, where C ≥ e is a given constant, if for every A, B ⊂ G 0 , either This property is satisfied for a random graph in G(m, d), as can be seen, for example, in work of Broder-Frieze-Suen-Upfal. For every ξ > 0 there exists C = C(θ, ξ) > e so that with probability at least 1 − o(m −ξ ), G has (θ, C)-controlled edge density.
Remark 7.3. The lemma in [BFSU99] is stated for θ > d/d min sufficiently large.
A reading of the proof shows that one can take any θ ≥ 2d/d min and any C ≥ 100θ + 100ξ. However, considering Definition 7.1, this then implies that the lemma holds for θ ≥ d/d min , at a cost of doubling C.
Proposition 7.4. If G ∈ G(m, d) has minimum degree d min , maximum degree d = d max , and (θ, C)-controlled edge density, then there exists C ′ = C ′ (θ, C) so that for all x ∈ T p,d,R (G 0 ), Together with Lemma 7.2, this proposition immediately implies Proposition 5.2. The remainder of this section consists of the proof of Proposition 7.4. Given x ∈ T p,d,R (G 0 ) ⊂ R m , the set of vertices G 0 splits into blocks as follows. For i > 0, let Those vertices with x u = 0 contribute nothing to X h x , and so may be ignored. Whenever x u = 0, |x u | p−1 ≥ ǫd 1/p /d i m 1/q ≥ ǫ/(dm) 1/q , and so u ∈ A i for some i ≥ 1.
Consider the function E ij (G) = E Ai,Aj (G) defined as the number of unoriented edges between A i and A j . With θ ≥ d/d min as above, let µ ij = a i a j θd/m.
If e ∈ E h , V(e) = {u, v}, with u ∈ A i and v ∈ A j , then where i ∧ j denotes the minimum of i and j.
We now bound X h x a.a.s. as follows.
Let us call the first of these terms X ha , and the second X hb . We bound each of these in turn.
Lemma 7.7. We have We now split this sum into five terms (cf. [BFSU99]), with C ′ b = D 1 ⊔ D 2 ⊔ D 3 ⊔ D 4 ⊔ D 5 , where D l denotes the subset of C ′ b satisfying (l), but not (l ′ ) for any l ′ < l. The parameter η > 0 will be optimised later.
Using these estimates, we see that Again, a geometric series argument shows that the two sums over i are bounded by 2d (p−1)η and 2pd η/(p−1) ≤ 2pd (p−1)η respectively. So
Lemmas 7.6 and 7.7 combine with (7.5) to complete the proof of Proposition 7.4. We have now completed the proof of Theorems 1.17 and 3.1.

Application to fixed point properties
Our main interest in estimates of the eigenvalues of the p-Laplacian resides in the following application. The result is proved using a slight modification of arguments of Bourdon [Bou12].
Proof. Assume that Γ acts by affine isometries on a Banach space V with L-bi-Lipschitz L p geometry above dimension m + 1, where L = (2 − 2ε) 1/2p . Following the terminology from [Bou12], we denote by E the set of all Γ-equivariant functions ψ : X 0 → V . We say that such a function is p-harmonic if it minimizes the energy, i.e.
is minimal among all values taken for functions in E, where Ξ 1 is a system of representatives of Γ\X 1 .
If inf ψ∈E E(ψ) = 0 and there exists a harmonic function then Γ has a fixed point and the argument is finished.
If either there exists no harmonic function or inf ψ∈E E(ψ) > 0 then Proposition 3.1 in [Bou12] implies that, up to replacing V with a rescaled ultralimit of itself, we can assume that there always exists a harmonic function ϕ and that E(ϕ) > 0. Note that all the hypotheses on V are stable by rescaled ultralimit.
An arbitrary vertex x has by hypothesis at most m neighbours. In particular ϕ(x) and ϕ(y) for y ∼ x span a subspace of dimension at most m + 1 hence, for L = (2 − 2ε) 1/2p , there exists an L-bi-Lipschitz map F x from a subspace U x containing ϕ(x) and ϕ(y) for y ∼ x to a space W x equal to an ℓ p n for some n ≥ m+1, or to ℓ p ∞ , or to some space L p (X, µ). Without loss of generality we may assume that F x (0) = 0.
Corollary 1.4, Proposition 2.4 and Lemma 4.1 in [Bou12] hold in full generality, in particular for the action of Γ on V . Proposition 2.4 applied to the harmonic function ϕ combined with Corollary 1.4 applied to F x • ϕ| L(x)0∪{x} imply that According to Lemma 4.1 in [Bou12] we may write where Ξ 0 is a system of representatives of Γ\X 0 and e xy denotes the edge of endpoints x, y. This and equation (8.1) imply that We have thus obtained that with the latter a strict inequality for E(ϕ) > 0, which gives a contradiction.
, has a fixed point.
Proof. Given an action on a space L p (X, µ) as described, a new norm can be defined on L p (X, µ), equivalent to the initial one, by the formula With respect to this new norm the action of Γ on L p (X) is isometric, and one can apply Theorem 1.16.

Fixed point properties in the triangular binomial model
In this section we prove Theorem 1.6, which finds fixed point properties with respect to actions on L p spaces, for random groups in the triangular binomial model.
Every finitely presented group has a finite triangular presentation, i.e. a presentation with all relators of length three. If Γ = S|R is a triangular finite presentation of a group, then Γ acts on a simplicial 2-complex X which is the Cayley complex. The link of a vertex in X is the graph L(S) with vertex set S ∪ S −1 and, for each relator of the form s x s y s z in R, edges (s −1 x , s y ), (s −1 y , s z ), and (s −1 z , s x ). Thus, the edges of L(S) decompose into three classes, corresponding to the order of appearance in the relators, and we decompose L(S) into three subgraphs L 1 (S), L 2 (S), L 3 (S), which each have the same vertex set as L(S), but only edges of the corresponding type.
Proof. Let C be the subspace of constant functions in R L0 . By (2.3), we have: We now show that adding a small number of edges to a graph cannot lower λ 1,p significantly.
Lemma 9.2. Let G and H be graphs with the same vertex set G 0 , and let G ∪ H denote the graph with vertex set G 0 and edge set G 1 ∪ H 1 .
If there exists ι > 0 so that for all u ∈ G 0 , val H (u) ≤ ι val G (u) then Proof. By (2.3), we have: We now follow [A LŚ15, Proof of Theorem 16] to describe the structure of link graphs for Cayley complexes of random groups in the model Γ(m, ρ) in terms of random graphs in a model G(2m, ρ ′ ). For the remaining relations, the probability that there is (at least) one edge between vertices u = v is ρ ′ = 1 − (1 − ρ) 4m−4 . Provided (2m) 2 (4m) 3 ρ 3 = o(1) there are no triple edges, and provided (2m)(2m) 2 (2m) 2 ρ 4 = o(1) no double edges share an endpoint, and so one matching deals with possible multiple edges.
We can now prove Theorem 1.6, in fact we will show the following stronger result.
Theorem 9.4. For any ε > 0 there exists C > 0 so that for any function f : m 2 , a.a.s. a random triangular group in the model Γ(m, ρ) has the property F L p (2−2ε) 1/2p for every p ∈ 2, 1 C (log f (m)/ log log f (m)) 1/2 . In particular, for a random triangular group, every affine action on an L p space that is (2 − 2ε) 1/2p -Lipschitz has a fixed point.
Moreover, if f (m)/ log m → ∞ as m → ∞, we can choose C independent of ǫ.
Remark 9.5. Observe that in the case of density d > 1/3 we have f (m) = m δ for some δ > 0, and that we get F L p in a range [2, 1 C (log m/ log log m) 1/2 ]. In the borderline case of f (m) = C log(m), we get F L p in the smaller, but still growing, range of [2, 1 C (log log m/ log log log m) 1/2 ]. Remark 9.6. As the random triangular groups are hyperbolic, this theorem is to be compared with the conjecture of Y. Shalom, stating that every Gromov hyperbolic group has an affine uniformly Lipschitz action on a Hilbert space that is proper [Sha01].
Proof. First we can assume ρ ≤ m δ /m 2 , hence f (m) ≤ m δ , for some δ < 1 4 . Since F L p is preserved by taking quotients, this case suffices.
The Mean Value Theorem implies that and that for m large enough For Γ ∈ Γ(m, ρ) by Proposition 9.3 with probability 1 − O(m −1+4δ /m 2 ), L 1 (S) is the union of a graph G 1 ∈ G(2m, ρ ′ ) with a matching. Theorem 1.17 gives that for C large enough there exists C ′ so that a.a.s.

Monotonicity and conformal dimension
In this section we discuss two consequences of Theorem 9.4: First, we use monotonicity to show a corresponding statement in the triangular density model. Second, we show conformal dimension bounds for random groups in both these models, which in turn shed light on the quasi-isometry types of such groups.
10.1. Monotonicity. We begin by comparing the triangular binomial/density models and the Gromov binomial/density models using standard monotonicity results for random structures, following [J LR00, Section 1.4].
A property of a group presentation is increasing if it is preserved by adding relations, and it is decreasing if it is preserved by deleting relations; it is monotone if it is either increasing or decreasing. For example, property F L p and being finite are both monotone (increasing) properties, and being infinite is a monotone (decreasing) property.
Let M(m, f (m)) be the triangular density model where we choose f (m) cyclically reduced relators of length three when we have m generators; the case of f (m) = (2m − 1) 3d , d ∈ (0, 1), is the usual triangular density model. In particular, if for all d > d 0 a random group in Γ(m, ρ), ρ = m d /m 3 , has P a.a.s. then for all d > d 0 a random group in M(m, d) has P a.a.s.
Let G(k, l, f (l)) be the Gromov density model where we choose f (l) cyclically reduced relators of length l. For example, given d ∈ (0, 1), G(k, l, (2k − 1) dl ) is the usual Gromov density model at density d.
In particular, if for all d > d 0 a random group in G(k, l, ρ), ρ = (2k − 1) −(1−d)l has P a.a.s., then for all d > d 0 a random group in G(k, l, d) has P a.a.s.
Propositions 10.1 and 10.2 both follow immediately from [J LR00, Proposition 1.13]. Similar statements to translate a.a.s. properties from the density models back to the binomial models follow from [J LR00, Proposition 1.12], but we do not need these here.
Having F L p L is a monotone property, so an immediate consequence of Proposition 10.1 and Theorem 9.4 is the following.
Corollary 1.7. For any fixed density d > 1/3 there exists C > 0 so that for every ε > 0 a.a.s. a random group in the triangular density model M(m, d) has F L p (2−2ε) 1/2p for every p ∈ 2, C(log m/ log log m) 1/2 . In particular, a.a.s. we have F L p for all p in this range.
10.2. Conformal dimension bounds. As discussed in the introduction, the conformal dimension Confdim(∂ ∞ Γ) of the boundary of a hyperbolic group Γ is an analytically defined quasi-isometry invariant of Γ. In this section we find the following bounds on conformal dimension in the triangular density model. (Similar bounds hold in the triangular binomial model.) Theorem 1.11. For any density d ∈ ( 1 3 , 1 2 ), there exists C > 0 so that a.a.s. Γ ∈ M(m, d) is hyperbolic, and satisfies The same holds for Γ(m, ρ) with ρ = m 3(d−1)+o(1) . In particular, as m → ∞, the quasi-isometry class of Γ keeps changing.
The connection between conformal dimension and property F L p is given by the following result. Proof. Indeed, all relators have length three, so one can take .
This in turn yields our desired upper bound for the conformal dimension.
Proposition 10.6. If ρ = m 3(d−1)+o(1) for some d < 1 2 , then a.a.s. Γ ∈ Γ(m, ρ) has Each of these steps also applies to the model M(m, d) for d ∈ ( 1 3 , 1 2 ), so Theorem 1.11 is proved. 11. Multi-partite (random) graphs and bounding λ 1,p In the remainder of this paper, we wish to extend some of our results from the triangular models of random groups to the Gromov models. This involves quite a few technicalities when done carefully; see for example Kotowski-Kotowski [KK13]. One approach they take to showing Property (T) for groups in the Gromov density model is to use an auxiliary bipartite model. Unfortunately Proposition 11.6 implies that this strategy does not work for F L p with large p. Instead we shall use a different auxilliary model based on complete multi-partite graphs.
In this section, we bound λ 1,p for random multi-partite graphs, and in Section 12 we apply it to random groups in the Gromov models.
11.1. Complete multi-partite graphs. Consider a complete k-partite graph with k independent sets of vertices, each of M vertices, and m = kM the total number of vertices. We denote such a graph by K k×M . These are particular cases of Turán graphs. In this subsection we find bounds on λ 1,p (K k×M ).
When M = 1 we have the complete graph on m vertices, and the following theorem gives the value of λ 1,p in this case.
Theorem 11.1 (Corollary 2, §9, in [Amg03]). If p > 2 then the smallest positive eigenvalue of the p-Laplacian for the complete graph K m with m vertices is Using this, we can prove the following estimate Theorem 11.2. If p > 2, k, M ≥ 2 then the smallest positive eigenvalue of the p-Laplacian for the graph K k×M satisfies where m = kM .
Proof. In what follows we fix the two arbitrary integers k ≥ 2 and M ≥ 2. Let V be its set of vertices and let V = V 1 ⊔ · · · ⊔ V k be the partition into k sets containing M vertices. Let x be a non-constant function in R V such that u∈V {x u } p−1 d u = 0, x p p,d = u∈V (1 − 1/k)m|x u | p = 1. We denote by dx the total derivative of x with respect to the set of edges in the graph K k×M , and by d c x the total derivative of x with respect to the set of edges in the complete graph K m .
The upper bound is trivial: choose any such x where x is zero on V i for all i ≥ 2, and then by (2.5), λ 1,p (K k×M ) ≤ dx p p = 1. In the remainder of the proof we show the lower bound for arbitrary such x.
Let a ∈ V be the vertex such that v∼a |x v − x a | p takes the minimal value among all the vertices in V . By summing over every edge twice, it follows that Without loss of generality we may assume that a ∈ V 1 , which means that the sum can be re-written as Hölder's inequality implies that for any two positive numbers α, β, Therefore for every v, w ∈ V i we can write, using the triangle inequality and the inequality above, that We may therefore write that Without loss of generality we may assume that b ∈ V k , and an argument as above implies that The inequalities (11.3) and (11.4) imply that Let y be the function y = ((1−1/k)/(1−1/m)) 1/p x, so that u∈V (m−1)|y u | p = 1. Since y is an eligible function for K m in (2.5), by Theorem 11.1 we have that It follows that For p = 2, we can do better; this will be useful when showing property (T ) later.
Proof. Denote the values of a function x on the vertices of K k×M by x i,u for 1 ≤ i ≤ k, 1 ≤ u ≤ M , with the first subscript indicating the partition into k sets. Then, if x ∈ S 2,d ((K k×M ) 0 ), by (2.5) we have and equality is attained for any function x with u x i,u = 0 for all 1 ≤ i ≤ k.
Given this bound, one might wonder about the sharpness of Theorem 11.2. However, the following proposition shows that, at least in the case of k = 2, the theorem's estimate is fairly accurate.
Proposition 11.6. For any fixed p > 2, as M → ∞ we have Proof. The lower bound of (1/2 − o(1))/2 p follows from Theorem 11.2 above. We use an explicit function to give an upper bound for λ 1,p (K 2×m ) via (2.4). We define a function x on the 2M vertices of K 2×M which depends on two parameters δ, t ∈ (0, 1). On δM of the left (respectively right) vertices, let x take the value 1 (resp. −1). On the remaining (1 − δ)M of the left (resp. right) vertices, let x take the value −t (resp. t). This function x satisfies the conditions of (2.4), so we can use it to give an upper bound for λ 1,p (K 2×M ). We do so with the (near optimal) choices of t = 1/5, δ = t p/2 = 5 −p/2 . (The error caused by rounding δM to the nearest integer disappears as M → ∞.) = (1 + o(1)) 1 2 · 5 −p/2 2 p + ( 4 5 ) p + 1 2 2 p 5 −p/2 ≤ (1 + o(1)) 2 √ 5 p 11.2. Multi-partite random graphs. We can view random graphs G(m, ρ) as arising from taking the complete graph K m and keeping each edge with probability ρ. The following model is defined analogously using K k×M as the base graph.
Definition 11.7. For k ≥ 2, M ∈ N and ρ ∈ [0, 1] a random k-partite graph G in the model G k (M, ρ) is found by taking the graph K k×M and keeping each edge with probability ρ. A property holds a.a.s. if it holds with probability → 1 as M → ∞.
In this model, we show the following two bounds on λ 1,p at slightly different ranges of ρ.
Recall that in K k×M the vertex set splits as V = k i=1 V i with an edge joining u ∈ V i to v ∈ V j if and only if i = j. For a graph G ∈ G k (M, ρ), and u ∈ V, 1 ≤ i ≤ k, let d u,i be the number of edges with one endpoint at u and the other endpoint in V i . So the degree of u is d u = k i=1 d u,i , and d u,i = 0 when u ∈ V i . Let D = D(G) = (d u,i ) u,i be the degree matrix of G.
We call a matrix D = (d u,i ) with integer entries an admissible degree matrix if for i = j, u∈Vi d u,j = v∈Vj d v,i ; we denote by ∆ i,j the common value of the two sums.
Given an admissible degree matrix D, we define a random graph model G k (M, D) as follows. We attach to each u ∈ V i a collection of d u half-edges, d u,j of which "point towards" V j for each j, and then for each i = j we join the collections of ∆ i,j half-edges pointing to each other by a random matching.
In the particular case of k = 2, this is just a random bipartite graph with specified degrees.
Given G ∈ G k (M, ρ), let Y u,i be the random variable which is d u,i , and let Y u = val G (u) = i Y u,i . These satisfy EY u,i = M ρ and EY u = (k − 1)M ρ =:d.
Lemma 11.10. Given ι = 10 log(M k)/M ρ, a.a.s. for all u, i, Proof. As in the proof of Lemma 3.3, we have M k(k −1) binomial random variables with expected value M ρ, so the probability that the claim fails is at most for all y ∈ S p,d (G 0 ), Z y (G) is at least (11.14) As before, we can assume that (1 + 2(ǫθ) 1/(p−1) ) p−1 ≤ 2. We now conclude the proofs of Theorems 11.8 and 11.9 using (11.14) and (2.5).
The bound for all p ′ in the given range follows from the argument of subsection 5.2.
Proof of Theorem 11.9. We estimate the terms in (11.14). Since k = o(M δ/2 ), Likewise, Thus (11.14) is bounded from below by which simplifies to give the claimed bound. 11.3. Expectation of Z. For each i = j, let V i→j be the collection of ∆ i,j endpoints of half-edges from V i pointing towards V j . Given a ∈ V i→j and b ∈ V j→i , let I a,b (G) be the random variable which is 1 or 0 according to whether a and b are matched in G or not. For a ∈ V i→j , denote by v(a) ∈ V i the other endpoint of the half-edge ending at a. Then Thus we can use the definition of λ 1,p (K k×M ) in (2.4) to find (11.17) Using the Mean Value Theorem we have for (1 + ι) 1/(p−1) − 1 and 1 − (1 − ι) 1/(p−1) are both at most 2ι/(p − 1), for ι < 1/10. Consequently |x ′ u | p−1 ≤ (1 + 2ι/(p − 1)) p−1 |x u | p−1 ≤ e 2ι |x u | p−1 ≤ 2|x u | p−1 . We also require the following inequality, a straightforward consequence of the Mean Value Theorem (see also [Mat97,Lemma 4]): For any p ≥ 1 and a, b ∈ R we have |{a} p − {b} p | ≤ p|a − b| |a| p−1 + |b| p−1 .
Proof. As in the proof of Proposition 6.6, we order the vertices of each V i→j and define a filter (F t ) on G k (M, D) as follows: first expose the edge connected to the first vertex of V 1→2 , then the second, and so on, then continue with V 1→3 , . . . , V 1→k , V 2→3 etc. Let F t be the σ-algebra generated by the first t exposed edges. As before, let S t = E(X l x |F t ), so S 0 = E(X l x ) and S E = X l x . For edges e which contribute to X l x , we have |ℜ(e)| ≤ p2 p ℜ(e) ≤ p2 p d β /dm, thus the same argument as before gives |S t (G) − S t−1 (G)| ≤ p2 p+2 d β /dm. Azuma's inequality tells us that |X l x − EX l x | has the desired lower bound with probability less than The desired inequality follows from Proposition 4.5 (compare (6.9)).
11.6. Controlled edge density. In light of Proposition 7.4, to bound X h x it suffices to show that G ∈ G k (M, D) has controlled edge density.

Fixed points for random groups in Gromov's density model
In this section we use the bounds on λ 1,p for random multi-partite graphs to show the following fixed point properties for the Gromov binomial and density models (see Definition 1.8).
The arguments in this section owe a debt to those of [A LŚ15] and [KK13], though our approach to Gromov's density model is new even for property (T), and gives new results at density d = 1/3, see Theorem 12.6 and Corollary 12.7. It is reasonable to expect that the dependence of k on p is unnecessary in Theorem 12.1, however our methods are not at present able to avoid this obstacle.
Suppose we are given l ∈ N that is a multiple of 3. Let W l/3 be the collection of all reduced words in A of length l, so |W l/3 | = 2k(2k − 1) l/3−1 . The map w → w −1 on W l/3 is a fixed point free involution. Choose a set S of size 1 2 |W l/3 | and a injection φ : S → W so that φ(S) is a collection of orbit representatives of this involution.
Proof. For any reduced word ab of length 2 in the generators A ∪ A −1 , we can find a word w of length l/3 − 1 so that aw and b −1 w are both reduced. Thus there are generators s, t ∈ S ∪ S −1 so that ab = (aw)(w −1 b) = φ(s)φ(t) = φ(st).
Therefore for g ∈ Γ, if g has even length it lies in φ(Γ), and if g has odd length it lies in one of the finitely many cosets aφ(Γ), a ∈ A ∪ A −1 .
So to show that Γ has F L p is suffices to show thatΓ has the same property, and for this we show that the link graph L(S) ofΓ has λ 1,p (L(S)) > 1/2. As in the proof of Theorem 9.4, we split L(S) as a union of three graphs L(S) = L 1 (S) ∪ L 2 (S) ∪ L 3 (S) where for each relation stu ∈R we put the edge (s −1 , t) in L 1 (S), the edge (t −1 , u) in L 2 (S), and the edge (u −1 , s) in L 3 (S).
For each a ∈ A ∪ A −1 , let S a be the subset of S consisting of generators s so that φ(s) ∈ W l/3 has initial letter a; S a has size M = (2k − 1) l/3−1 . Observe that st can begin a relation stu ∈R if and only if φ(s)φ(t) is a reduced word in A , which holds exactly when φ(s) −1 = φ(s −1 ) and φ(t) have different initial letters. In other words, s −1 and t lie in different sets of the partition S = a∈A∪A −1 S a . We now show that each L i (S) is the union of a random 2k-partite graph with two matchings.
We require a count on the number of ways to complete a cyclically reduced word.
Proof. We define an auxiliary multigraph K with the same vertex set as K 2k×M . For s −1 ∈ S a ⊂ S and t ∈ S b ⊂ S with a = b, there are q l/3 or q l/3 + 1 possible ways to complete φ(s)φ(t) to a cyclically reduced word φ(s)φ(t)φ(u) of length l, and the same number of ways to complete φ(t −1 )φ(s −1 ), depending on the final letters of φ(s −1 ) and φ(t). Accordingly, add 2q l/3 or 2q l/3 + 2 edges to K between s −1 and t.
Then L 1 (S) can be viewed as the random graph obtained from K by retaining each edge with probability ρ ′ .
For each pair of vertices in K connected by 2q l/3 + 2 edges, delete two edges and call the resulting graphK; let D be the collection of deleted edges. Let L 1 (S) = L ∪D whereL is the portion of L 1 (S) coming fromK andD the portion coming from D.
So L 1 (S) is the union of a graph L ′ ∈ G 2k (M, ρ ′ ), a matching coming from the double edges ofL and the matchingD.
We are now able to show F L p for random groups in the Gromov binomial model, and hence the Gromov density model as well.
Proof of Theorem 12.1. By Proposition 10.2 it suffices to consider the model G(k, l, ρ) for ρ(l) = (2k − 1) −(1−d)l . By Proposition 12.4, a.a.s. Γ ∈ G(k, l, ρ) is, up to finite index, the quotient of a groupΓ whose link graph is the union of three graphs each consisting of a graph G ∈ G 2k (M, ρ ′ ) and two matchings.
Finally, we use our results to give a new proof of Kazhdan's property (T) for random groups in Gromov's density model at d > 1/3, and moreover give new information at d = 1/3.