Homomorphic Images of Locally Compact Groups Acting on Trees and Buildings

TL;DR

This paper investigates the structure of totally disconnected locally compact groups acting on trees and buildings, establishing conditions under which their continuous homomorphic images are guaranteed to be closed, inspired by Cartan decompositions of Lie groups.

Contribution

It introduces analogues of Cartan decompositions for these groups and proves that many such groups have all continuous homomorphic images closed, advancing understanding of their algebraic and topological properties.

Findings

Many groups acting on trees and buildings have all continuous homomorphic images closed.

The paper develops Cartan decomposition analogues for totally disconnected locally compact groups.

These decompositions help characterize the structure and homomorphic images of such groups.

Abstract

We study analogues of Cartan decompositions of Lie groups for totally disconnected locally compact groups. It is shown using these decompositions that a large class of totally disconnected locally compact groups acting on trees and buildings have the property that every continuous homomorphic image of the group is closed.