Commentarii Mathematici Helvetici


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Volume 93, Issue 2, 2018, pp. 291–333
DOI: 10.4171/CMH/435

Published online: 2018-05-31

Random walks and boundaries of CAT(0) cubical complexes

Talia Fernós[1], Jean Lécureux[2] and Frédéric Mathéus[3]

(1) University of North Carolina, Greensboro, USA
(2) Université Paris-Sud 11, Orsay, France
(3) Université de Bretagne-Sud, Vannes, France

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.

Keywords: CAT(0) cube complexes, Roller boundary, visual boundary, random walks, stationary measure, drift

Fernós Talia, Lécureux Jean, Mathéus Frédéric: Random walks and boundaries of CAT(0) cubical complexes. Comment. Math. Helv. 93 (2018), 291-333. doi: 10.4171/CMH/435