112 research outputs found
Vanishing of the first reduced cohomology with values in an -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
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
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
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
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 has property PL if every isometric -action
either has bounded orbits or is (metrically) proper. For , say that
has property if the same alternative holds for the smaller class of
affine isometric actions on -spaces. We explore properties PL and
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 be a surface group of genus . It is known that the
canonical central extension and the direct product
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 -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
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