Contact graphs, boundaries, and a central limit theorem for $\mathrm{CAT}(0)$ cubical complexes

Abstract

Let $X$ be a nonelementary $\mathrm{CAT}(0)$ cubical complex. We prove that if $X$ is essential and irreducible, then the contact graph of $X$ (introduced by Hagen (2014)) is unbounded and its boundary is homeomorphic to the regular boundary of $X$ (defined by Fernós (2018) and Kar–Sageev (2016)). 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 let $(Z_n)$ be 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 ($A={\lim }_{n\to \infty }\frac{d(Z_{n}o,o)}{n}$ is the drift of the random walk, and $o\in X$ is an arbitrary basepoint).