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

Pierre-Emmanuel Caprace,Nicolas Monod

doi:10.5802/jep.26

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 ≥ 2 is a Bruhat–Tits building if and only if its automorphism group acts cocompactly and chamber-transitively at infinity.

Highlights

Quand le vent est au rire, quand le vent est au blé Quand le vent est au Sud, écoutez-le chanter Le plat pays qui est le mien.(Jacques Brel, Le plat pays, 1962)The meeting ground between non-positive curvature and amenability is shaped by flatness
We provide 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 2 is a Bruhat–Tits building if and only if its automorphism group acts cocompactly and chamber-transitively at infinity
This principle emerged in the 1970s for Riemannian geometry [5], [20], [27] and culminated in 1998 as a definitive metric statement: The flat Euclidean spaces are the only geodesically complete locally compact CAT(0) spaces admitting a proper cocompact isometric action of an amenable discrete group [2, Cor

Summary

Introduction

Quand le vent est au rire, quand le vent est au blé Quand le vent est au Sud, écoutez-le chanter Le plat pays qui est le mien. The main objective of this article is to establish the classification of geodesically complete CAT(0) spaces without global fixed point at infinity that admit a cocompact amenable group of isometries: they are all products of flats, symmetric spaces, Bruhat–Tits buildings and trees (Theorem B below). A more fancy-clad construction gives a continuous deformation of the classical hyperbolic plane [32] The latter provides a one-parameter family of (homothety classes of) nongeodesically complete proper cocompact minimal CAT(−1) spaces whose isometry group is PGL2(R). We focus on proper CAT(0) spaces admitting a cocompact isometric action of a totally disconnected amenable locally compact group This part of the work (which represents about half of the proof of Theorem A) is valid in full generality: global fixed points at infinity are allowed. We are grateful to Richard Weiss for a useful comment on the classification of locally finite Bruhat–Tits buildings and to the anonymous referee for useful remarks

Geometric preliminaries

Spherical buildings

Boundary points with a cocompact stabiliser

Building buildings by means of means