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 ${L}^{\infty}$-cocycles for characteristic classes.

We prove that the natural map ${H}_{\text{b}}^{2}\left(\Gamma \right)\to {H}^{2}\left(\Gamma \right)$ from bounded to usual cohomology
is injective if $\Gamma $ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for $\Gamma $:
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 ${H}^{*}\text{b}\left(\Gamma \right)$ to the continuous bounded cohomology of the ambient group...

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

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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