Random walks and boundaries of CAT(0) cubical complexes

Abstract

We show under weak hypotheses that the pushforward {Z n o} 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.

Cross ratios and cubulations of hyperbolic groups

“…We will write g ± X ∈ ∂ cnt X when it is necessary to specify the cube complex. The next proposition follows from Lemmas 2.9 and 4.7 in [4], although the main ingredients are actually from [31]. The same result holds for any finite collection of (irreducible, essential, cocompact) cubulations.…”

“…We stress that our definition of contracting point is not equivalent to the one in Remark 6.7 of [31]; in fact, our notion is weaker.…”

“…According to an unpublished result of U. Bader and D. Guralnik (and [18,31]), the identity of X extends to a homeomorphism between ∂ X and the horofunction boundary of (X , d) (at least when X is proper, see Remark 6.19 in [31]). However, the characterisation of X in terms of ultrafilters additionally provides a natural structure of median algebra on X , corresponding to the map…”

“…The following notion appeared in Definition 7.3 in [28] and Definition 5.8 in [31]; also see Proposition 7.5 in [28]. Definition 2.6 Let X be irreducible.…”

“…Given subsets U ⊆ W (X ) and V ⊆ W (Y ), we denote by U ⊆ H (X ) and V ⊆ H (Y ) the collections of halfspaces bounded, respectively, by the elements of U and V. Definition 4. 31 We say that subsets U ⊆ W (X ) and V ⊆ W (Y ) are well-paired if, for every h ∈ H (X ) and k ∈ H (Y ), we have:…”