On the vanishing of reduced 1-cohomology for Banach representations

A theorem of Delorme states that every unitary representation of a connected solvable Lie group with nontrivial reduced first cohomology has a nonzero 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 Prüfer 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.