We provide several new characterizations of Property A for bounded degree graphs. In particular, we show that (X,d) has Property A if and only if there is a proper gauge ω such that the Lipschitz free space LF(X,ω∘ d) is isomorphic to ℓ_1. As a consequence, all finitely generated groups with Property A admit proper uniformly Lipschitz affine actions on ℓ_1. Moreover, for groups with finite Nagata dimension, we obtain actions with compression exponent 1. This result applies to higher rank lattices, such as SL(3,ℤ). We also show that a countable discrete group coarsely embeds into L_1 if and only if it admits a proper uniformly Lipschitz affine action on a subspace of L_1.

For pairs (G,H), where G is a locally compact group and H is a closed subgroup of G, we introduce analogous versions of the full group C^*-algebra and the Fourier–Stieltjes algebra for uniformly bounded representations. We use these objects to characterise relative property (T) in this more general setting in terms of Kazhdan projections and invariant means. We also show that, for semidirect products of the form G=H⋉ N, where N is a nilpotent group, relative property (T) for the pair (G,N) is equivalent to its uniformly bounded version. We first prove this result when N is abelian, and then proceed by studying the behaviour of relative property (T) under quotients by central subgroups, thereby extending a theorem of Serre to the uniformly bounded setting.

We exhibit an obstruction for groups with Relative Property (T) to act on the real line by bi-Lipschitz homeomorphisms. This condition is expressed in terms of the Lipschitz and Kazhdan constants associated to finite generating subsets. As an application, we obtain an explicit lower bound for the Lipschitz constants associated to actions of the semidirect product F2 x Z2. We also obtain an upper bound for the Kazhdan constants of pairs of orderable groups, depending only on the cardinal of the generating subset.

We prove an extension property for M_d-multipliers from a subgroup to the ambient group, showing that M_d+1(G) is strictly contained in M_d(G) whenever G contains a free subgroup. Another consequence of this result is the stability of the M_d-approximation property under group extensions. We also show that Baumslag-Solitar groups are M_d-weakly amenable with (BS(m,n),d)=1 for all d≥ 2. Finally, we show that, for simple Lie groups with finite centre, M_d-weak amenability is equivalent to weak amenability, and we provide some estimates on the constants (G,d).

We define the notion of almost invariant conditionally negative definite kernel and use it to give a characterisation of groups admitting a proper uniformly Lipschitz affine action on a subspace of an $L^1$ space. We show that this condition is satisfied by groups acting properly on products of quasi-trees, weakly amenable groups with Cowling-Haagerup constant 1, and a-TTT-menable groups.

We characterise Geometric Property (T) by the existence of a certain projection in the maximal uniform Roe algebra , extending the notion of Kazhdan projection for groups to the realm of metric spaces. We also describe this projection in terms of the decomposition of the metric space into coarsely connected components.

We present an introduction to weak amenability for locally compact groups, and a survey of some of the most important results regarding this property.

We show that the Lipschitz free space of a countable simplicial quasi-tree is isomorphic to ℓ^1. As a consequence, every finitely generated group with Property (QT) of Bestvina–Bromberg–Fujiwara has a proper uniformly Lipschitz affine action on ℓ^1 with quasi-isometrically embedded orbits. We also show that 3-manifold groups admit proper uniformly Lipschitz affine actions on ℓ^1.

We show that Property (TTT) is an obstruction to weak amenability with Cowling–Haagerup constant 1. More precisely, if G is a countable group and H is an infinite subgroup of G such that the pair (G, H) has relative Property (TTT) , then the weak Haagerup constant Λ_WH(G) is strictly greater than 1. We apply this result to some semidirect products and lattices in higher rank algebraic groups.

We define a strengthening of the Haagerup–Kraus approximation property by means of the subalgebras of Herz–Schur multipliers M d ( G ) M_d(G) ( d ≥ 2 d\geq 2 ) introduced by Pisier. We show that unitarisable groups satisfying this property for all d ≥ 2 d\geq 2 are amenable. Moreover, we show that groups acting properly on finite-dimensional CAT(0) cube complexes satisfy M d M_d -AP for all d ≥ 2 d\geq 2 . We also give examples of non-weakly amenable groups satisfying M d M_d -AP for all d ≥ 2 d\geq 2 .

We show that every hyperbolic group has a proper uniformly Lipschitz affine action on a subspace of an $L^1$ space. We also prove that every acylindrically hyperbolic group has a uniformly Lipschitz affine action on such a space with unbounded orbits. Our main tools are the $\mathbb{Q}$-bicombings on hyperbolic groups constructed by Mineyev and the characterisation of acylindrical hyperbolicity in terms of actions on quasi-trees by Balasubramanya.

Abstract We show that if G is an amenable group and H is a hyperbolic group, then the free product $G\ast H$ is weakly amenable. A key ingredient in the proof is the fact that $G\ast H$ is orbit equivalent to $\mathbb{Z}\ast H$ .

For every c≥ 1, we define a strengthening of Kazhdan's Property (T) by considering uniformly bounded representations π with fixed bound |π|≤ c. We carry out a systematic study of this property and show that it can be characterised by the weak*-continuity of the unique invariant mean on a suitable space of coefficients. For countable groups, we prove that the family of properties thus obtained yield an invariant at the von Neumann algebra level. Moreover, by focusing on certain representations of rank 1 Lie groups, we show that Sp(n,1) and F_4,-20 admit proper uniformly Lipschitz affine actions on Hilbert spaces.

We give a new proof of a classical result which provides a one-to-one correspondence between positive definite radial kernels on a homogeneous tree and finite Borel measures on the interval $[-1,1]$. Our methods allow us to find a new characterisation in terms of positive trace-class operators on $\ell_2$. Furthermore, we extend both characterisations to finite products of homogeneous trees. The proof relies on a formula for the norm of radial Schur multipliers, in the spirit of Haagerup--Steenstrup--Szwarc, and a variation of the Hamburger moment problem.

We give a characterisation of radial Schur multipliers on finite products of trees. The equivalent condition is that a certain generalised Hankel matrix involving the discrete derivatives of the radial function is a trace class operator. This extends Haagerup, Steenstrup and Szwarc's result for trees. The same condition can be expressed in terms of Besov spaces on the torus. We also prove a similar result for products of hyperbolic graphs, and provide a sufficient condition for a function to define a radial Schur multiplier on a finite-dimensional CAT(0) cube complex.

We prove that, for any 1 < p < infinity, the groups SL(3,R) and Sp(2, R) do not have the p-approximation property of An, Lee and Ruan, which implies in particular that they are not p-weakly amenable. It follows that the same holds for any connected simple Lie group with finite center and real rank greater than 1, as well as for any lattice in it. This extends Haagerup and de Laat's result for the AP, which in this language corresponds to the case p = 2. (C) 2017 Elsevier Inc. All rights reserved.

In this paper we are interested on the existence of ground state solutions for fractional field equations of the form {[ (I - Δ)^αu = f(x, u) in IR^N,; u > 0 in IR^N, lim_|x| →∞u(x) = 0, ]. where α∈ (0,1) and f is an appropriate super-linear sub-critical nonlinearity. We prove regularity, exponential decay and symmetry properties for these solutions. We also prove the existence of infinitely many bound states and, through a non-local Pohozaev identity, we prove nonexistence results in the supercritical case.