Vanishing of the first reduced cohomology with values in an LpL^p-representation

We prove that the first reduced cohomology with values in a mixing Lp-representation, p larger than 1, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group has no first reduced lp-cohomology. As a byproduct, we prove a conjecture by Pansu. Namely, the first reduced Lp-cohomology on homogeneous, closed at infinity, Riemannian manifolds vanishes. We also prove that a Gromov hyperbolic geodesic metric measure space with bounded geometry admitting a bi-Lipschitz embedded 3-regular tree has non-trivial first reduced Lp-cohomology for large enough p. Combining our results with those of Pansu, we characterize Gromov hyperbolic homogeneous manifolds: these are the ones having non-zero first reduced Lp-cohomology for some p larger than 1.Comment: 20 pages, correction: minor changes (introduction), corrections: we improved the redaction (in particular, adding more details to the proofs), and showed a more general statement for hyperbolic space

Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces

We characterize the asymptotic behaviour of the compression associated to a uniform embedding into some Lp-space for a large class of groups including connected Lie groups with exponential growth and word-hyperbolic finitely generated groups. In particular, the Hilbert compression rate of these groups is equal to 1. This also provides new and optimal estimates for the compression of a uniform embedding of the infinite 3-regular tree into some Lp-space. The main part of the paper is devoted to the explicit construction of affine isometric actions of amenable Lie groups on Lp-spaces whose compressions are asymptotically optimal. These constructions are based on an asymptotic lower bound of the Lp-isoperimetric profile inside balls. We compute this profile for all amenable connected Lie groups and for all finite p, providing new geometric invariants of these groups. We also relate the Hilbert compression rate with other asymptotic quantities such as volume growth and probability of return of random walks.Comment: 38 pages, modification: correct proof of the lower bound 2/3 of the compression of (Z \wr Z

Large scale Sobolev inequalities on metric measure spaces and applications

We introduce a notion of "gradient at a given scale" of functions defined on a metric measure space. We then use it to define Sobolev inequalities at large scale and we prove their invariance under large-scale equivalence (maps that generalize the quasi-isometries). We prove that for a Riemmanian manifold satisfying a local Poincare inequality, our notion of Sobolev inequalities at large scale is equivalent to its classical version. These notions provide a natural and efficient point of view to study the relations between the large time on-diagonal behavior of random walks and the isoperimetry of the space. Specializing our main result to locally compact groups, we obtain that the L^p-isoperimetric profile, for every p \in [1,\infty] is invariant under quasi-isometry between amenable unimodular compactly generated locally compact groups. A qualitative application of this new approach is a very general characterization of the existence of a spectral gap on a quasi-transitive measure space X, providing a natural point of view to understand this phenomenon.Comment: 43 page

Quantitative property A, Poincare inequalities, L^p-compression and L^p-distortion for metric measure spaces

We introduce a quantitative version of Property A in order to estimate the L^p-compressions of a metric measure space X. We obtain various estimates for spaces with sub-exponential volume growth. This quantitative property A also appears to be useful to yield upper bounds on the L^p-distortion of finite metric spaces. Namely, we obtain new optimal results for finite subsets of homogeneous Riemannian manifolds. We also introduce a general form of Poincare inequalities that provide constraints on compressions, and lower bounds on distortion. These inequalities are used to prove the optimality of some of our results.Comment: 26 page

Isoperimetric profile and random walks on locally compact solvable groups

We study a large class of amenable locally compact groups containing all solvable algebraic groups over a local field and their discrete subgroups. We show that the isoperimetric profile of these groups is in some sense optimal among amenable groups. We use this fact to compute the probability of return of symmetric random walks, and to derive various other geometric properties which are likely to be only satisfied by these groups.Comment: 23 page

On the L^p-distorsion of finite quotients of amenable groups

We study the L^p-distortion of finite quotients of amenable groups. In particular, for every number p larger or equal than 2, we prove that the l^p-distortion of the finite lamplighter group grows like (\log n)^{1/p}. We also give the asymptotic behavior of the l^p-distortion of finite quotients of certain metabelian polycyclic groups and of the solvable Baumslag-Solitar groups BS(m,1). The proofs are short and elementary.Comment: 8 page

Admitting a coarse embedding is not preserved under group extensions

We construct a finitely generated group which is an extension of two finitely generated groups coarsely embeddable into Hilbert space but which itself does not coarsely embed into Hilbert space. Our construction also provides a new infinite monster group: the first example of a finitely generated group that does not coarsely embed into Hilbert space and yet does not contain a weakly embedded expander.Comment: 15 pages; Proposition 3.3(v1) was modified following a comment of D. Sawicki; Theorem 2(v3) is new and gives an extension of finitely generated group

Locally compact groups with every isometric action bounded or proper

A locally compact group $G$ has property PL if every isometric $G$-action either has bounded orbits or is (metrically) proper. For $p>1$, say that $G$ has property $BP_{L^p}$ if the same alternative holds for the smaller class of affine isometric actions on $L^p$-spaces. We explore properties PL and $BP_{L^p}$ and prove that they are equivalent for some interesting classes of groups: abelian groups, amenable almost connected Lie groups, amenable linear algebraic groups over a local field of characteristic 0. The appendix by Corina Ciobotaru provides new examples of groups with property PL, including non-linear ones.Comment: 29 pages; with an appendix by Corina Ciobotar

Integrable measure equivalence and the central extension of surface groups

Let $\Gamma_g$ be a surface group of genus $g\geq 2$. It is known that the canonical central extension $\tilde{\Gamma}_g$ and the direct product $\Gamma_g\times \mathbb{Z}$ are quasi-isometric. It is also easy to see that they are measure equivalent. By contrast, in this paper, we prove that quasi-isometry and measure equivalence cannot be achieved "in a compatible way". More precisely, these two groups are not uniform (nor even integrable) measure equivalent. In particular, they cannot act continuously, properly and cocompactly by isometries on the same proper metric space, or equivalently they are not uniform lattices in a same locally compact group.Comment: 15 pages, no figures. In the previous version, we had overlooked a point in the proof of Theorem 1.1. This time we have strengthened this proof and we have added Theorem 1.

On the vanishing of reduced 1-cohomology for Banach representations

A theorem of Delorme states that every unitary representation of a connected Lie group with nontrivial reduced first cohomology has a finite-dimensional subrepresentation. More recently Shalom showed that such a property is inherited by cocompact lattices and stable under coarse equivalence among amenable countable discrete groups. We give a new geometric proof of Delorme's theorem which extends to a larger class of groups, including solvable $p$-adic algebraic groups, and finitely generated solvable groups with finite Pr\"ufer rank. Moreover all our results apply to isometric representations in a large class of Banach spaces, including reflexive Banach spaces. As applications, we obtain an ergodic theorem in for integrable cocycles, as well as a new proof of Bourgain's Theorem that the 3-regular tree does not embed quasi-isometrically into any superreflexive Banach space.Comment: 44 pages, no figur