A boundary map to the Roller boundary of a CAT(0) cube complex

I wrote this thesis at Heidelberg University at the Department of Mathematics & Computer Science. I was adviced by Professor Anna Wienhard.

Abstract

In this thesis we will present the construction of a so-called boundary map between the strong Γ-boundary B of a discrete, countable group Γ and the Roller boundary ∂X of a CAT(0) cube complex X on which Γ acts by automorphisms:

φ : B → ∂X.

We will see that this boundary map is measurable and Γ-equivariant almost everywhere. The existence was first proven by Chatterji, Fernós, and Iozzi under the further assumption that X is connected, locally countable and finite-dimensional and that Γ acts non-elementary on X. This thesis has an expository nature. We will give a brief introduction to CAT(0) cube complexes and then turn towards the Roller duality, which will lead us immediately to the Roller boundary. Additionally, we will explore group actions on CAT(0) cube complexes introducing the notions of non-elementarity and essentiality. Lastly, we will define ergodic group actions (with coefficients) and strong Γ-boundaries.

Building the thesis

This thesis was built on a linux system using the tex-live-full package. The packages I used make it necessary to use xelatex to build it (pdflatex will fail). The following commands should build the thesis:

git clone git@github.com:Emrys-Merlin/Master-Mathematics.git .
cd Master-Mathematics
xelatex Master.tex
biber Master
xelatex Master.tex

Afterwards you should find a Master.pdf file in your working directory.