Contact graphs, boundaries, and a central limit theorem for $\mathrm{CAT}(0)$ cubical complexes
Talia Fernós, Jean Lécureux, Frédéric Mathéus · Groups Geometry and Dynamics · 2024
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).