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

TL;DR

This paper studies the geometric and probabilistic properties of CAT(0) cubical complexes, establishing the unboundedness of their contact graphs, describing their boundaries, and proving a Central Limit Theorem for random walks on groups acting on these complexes.

Contribution

It demonstrates the unboundedness of contact graphs for essential irreducible CAT(0) cubical complexes, characterizes their boundaries, and proves a CLT for random walks on groups acting on such complexes.

Findings

Contact graphs are unbounded for essential irreducible complexes.

The boundary of the contact graph is homeomorphic to the regular boundary of the complex.

A Central Limit Theorem is established for random walks on groups acting on these 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).