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

Abstract:

In this thesis we will present the construction of a so-called boundary map between the strong \(\Gamma\)-boundary B of a discrete, countable group \(\Gamma\) and the Roller boundary \(\partial X\) of a CAT(0) cube complex \(X\) on which \(\Gamma\) acts by  automorphisms:
\[
\phi: B \rightarrow \partial X
\]
We will see that this boundary map is measurable and \(\Gamma\)-equivariant almost everywhere. The existence was first proven by Chatterji, Fernós, and Iozzi [CFI16] under  the further assumption that X is connected, locally countable and finite-dimensional and that \(\Gamma\) 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 \(\Gamma\)-boundaries.