On fixed points and uniformly convex spaces

TL;DR

This paper presents two elementary facts about actions on uniformly convex spaces and demonstrates their application in providing an alternative proof for the triviality of the first Lp-cohomology of higher rank simple Lie groups.

Contribution

It introduces two elementary facts about uniformly convex spaces and applies them to give an alternative proof of a known cohomology triviality result.

Findings

Two elementary facts about actions on uniformly convex spaces.

Alternative proof of the triviality of the first Lp-cohomology of higher rank simple Lie groups.

Abstract

The purpose of this note is to present two elementary, but useful, facts concerning actions on uniformly convex spaces. We demonstrate how each of them can be used in an alternative proof of the triviality of the first $L_p$-cohomology of higher rank simple Lie groups, proved in [BFGM].