Equicontinuous actions of semisimple groups

TL;DR

This paper investigates equicontinuous actions of semisimple groups, proving their properness and deriving applications like closedness of homomorphisms, ergodicity, and matrix coefficient decay.

Contribution

It establishes that equicontinuous actions of semisimple groups are universally closed and proper, with new applications in topology and representation theory.

Findings

Equicontinuous actions are universally closed and proper.

Continuous homomorphisms are closed.

Matrix coefficients vanish for reflexive and WAP representations.

Abstract

We study equicontinuous actions of semisimple groups and some generalizations. We prove that any such action is universally closed, and in particular proper. We derive various applications, both old and new, including closedness of continuous homomorphisms, nonexistence of weaker topologies, metric ergodicity of transitive actions and vanishing of matrix coefficients for reflexive (more generally: WAP) representations.