On the vanishing of reduced 1-cohomology for Banach representations

TL;DR

This paper provides a new geometric proof of Delorme's theorem on the vanishing of reduced first cohomology for certain Banach space representations, extending the result to broader classes of groups and applications.

Contribution

It introduces a geometric proof that extends Delorme's theorem to solvable p-adic groups and finitely generated solvable groups with finite Prüfer rank, applicable to Banach space representations.

Findings

Extended Delorme's theorem to new classes of groups including solvable p-adic groups.

Proved vanishing of reduced 1-cohomology for isometric Banach space representations.

Applied results to ergodic theorems and quasi-isometric embedding problems.

Abstract

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.