Contact Graphs, Boundaries, and a Central Limit Theorem for CAT(0) cubical complexes

Abstract

Let $X$ be a nonelementary CAT(0) cubical complex. We prove that if $X$ is essential and irreducible, then the contact graph of $X$ (introduced in \cite{Hagen}) is unbounded and its boundary is homeomorphic to the regular boundary of $X$ (defined in \cite{Fernos}, \cite{KarSageev}). Using this, we reformulate the Caprace-Sageev's Rank-Rigidity Theorem in terms of the action on the contact graph. Let $G$ be a group with a nonelementary action on $X$, and $(Z_n)$ a random walk corresponding to a generating probability measure on $G$ with finite second moment. Using this identification of the boundary of the contact graph, we prove a Central Limit Theorem for $(Z_n)$, namely that $\frac{d(Z_n o,o)-nA}{\sqrt n}$ converges in law to a non-degenerate Gaussian distribution (where $A=\lim \frac{d(Z_no,o)}{n}$ is the drift of the random walk, and $o\in X$ is an arbitrary basepoint).