Random Walks and Boundaries of CAT(0) Cubical complexes

TL;DR

This paper investigates the behavior of random walks on CAT(0) cubical complexes, demonstrating convergence to boundary points and introducing squeezing points to analyze boundary convergence, with implications for group actions.

Contribution

It introduces the concept of squeezing points and establishes convergence of random walks to boundary points under weak hypotheses, linking group actions to boundary dynamics.

Findings

Random walks converge to boundary points in CAT(0) cube complexes.

Introduction of squeezing points for boundary convergence analysis.

Any nonelementary action contains rank-1 hyperbolic elements.

Abstract

We show under weak hypotheses that the pushforward $\{Z_no\}$ of a random-walk to a CAT(0) cube complex converges to a point on the boundary. We introduce the notion of squeezing points, which allows us to consider the convergence in either the Roller boundary or the visual boundary, with the appropriate hypotheses. This study allows us to show that any nonelementary action necessarily contains regular elements, that is, elements that act as rank-1 hyperbolic isometries in each irreducible factor of the essential core.