  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 $p$-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.
