An indiscrete Bieberbach theorem: from amenable CAT(0) groups to Tits buildings

TL;DR

This paper classifies certain non-positively curved spaces with amenable symmetry groups, showing their boundaries are spherical buildings and characterizing when these spaces are products of well-known geometric structures.

Contribution

It establishes a classification of amenable CAT(0) spaces with no fixed point at infinity, linking their boundaries to spherical buildings and identifying conditions for these to be Bruhat--Tits buildings.

Findings

Visual boundary is a spherical building.

Spaces are products of flats, symmetric spaces, trees, and buildings.

Euclidean buildings are Bruhat--Tits if automorphism acts cocompactly and chamber-transitively.

Abstract

Non-positively curved spaces admitting a cocompact isometric action of an amenable group are investigated. A classification is established under the assumption that there is no global fixed point at infinity under the full isometry group. The visual boundary is then a spherical building. When the ambient space is geodesically complete, it must be a product of flats, symmetric spaces, biregular trees and Bruhat--Tits buildings. We provide moreover a sufficient condition for a spherical building arising as the visual boundary of a proper CAT(0) space to be Moufang, and deduce that an irreducible locally finite Euclidean building of dimension at least 2 is a Bruhat--Tits building if and only if its automorphism group acts cocompactly and chamber-transitively at infinity.