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Equivariant measurable liftings

Nicolas Monod — 2015

Fundamenta Mathematicae

We discuss equivariance for linear liftings of measurable functions. Existence is established when a transformation group acts amenably, as e.g. the Möbius group of the projective line. Since the general proof is very simple but not explicit, we also provide a much more explicit lifting for semisimple Lie groups acting on their Furstenberg boundary, using unrestricted Fatou convergence. This setting is relevant to L -cocycles for characteristic classes.

Bounded cohomology of lattices in higher rank Lie groups

Marc BurgerNicolas Monod — 1999

Journal of the European Mathematical Society

We prove that the natural map H b 2 ( Γ ) H 2 ( Γ ) from bounded to usual cohomology is injective if Γ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for Γ : the stable commutator length vanishes and any C 1 –action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating H * b ( Γ ) to the continuous bounded cohomology of the ambient group...

An obstruction to p -dimension

Nicolas MonodHenrik Densing Petersen — 2014

Annales de l’institut Fourier

Let G be any group containing an infinite elementary amenable subgroup and let 2 < p < . We construct an exhaustion of p G by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to p -dimension and gives an answer to a question of Gaboriau.

Amenable hyperbolic groups

Pierre-Emmanuel CapraceYves de CornulierNicolas MonodRomain Tessera — 2015

Journal of the European Mathematical Society

We give a complete characterization of the locally compact groups that are non elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semiregular trees acting doubly...

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