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A notion of geometric complexity and its application to topological rigidity
We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact…
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Amenable hyperbolic groups
We give a complete characterization of the locally compact groups that are nonelementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous…
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Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces
We characterize the possible asymptotic behaviors of the compression associated to a uniform embedding into some Lp-space, with 1 < p < ∞, for a large class of groups including connected Lie groups…
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Discrete groups with finite decomposition complexity
In [GTY] we introduced a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. In that article we proved the stable Borel conjecture for a…
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Metric sparsification and operator norm localization
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Isometric Group Actions on Hilbert Spaces: Growth of Cocycles
Abstract.We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following…
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Volume of spheres in doubling metric measured spaces and in groups of polynomial growth
(Volume de spheres dans les espaces metriques mesures doublants et dans les groupes a croissance polynomiale) Soit G un groupe localement compact, compactement engendre et U une partie compacte…
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Relative expanders
We exhibit a finitely generated group G and a sequence of finite index normal subgroups such that for every finite generating subset $${S\subseteq G}$$S⊆G, the sequence of finite Cayley graphs (G/Nn,…
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Large scale Sobolev inequalities on metric measure spaces and applications
For functions on a metric measure space, we introduce a notion of “gradient at a given scale”. This allows us to define Sobolev inequalities at a given scale. We prove that satisfying a Sobolev…
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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…
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