Lemma 3.18 was not stated correctly. This is fixed, and a minor adjustment to the beginning of the proof of Theorem 3.19 has been made as a result. Statements other than 3.18 do not need to change. I thank Abdul Zalloum for the correction. See also: arXiv:2004.01182 (this version differs from previous only by addition of the preceding link, at administrators' request)
For a CAT(0) cube complex $\mathbf X$, we define a simplicial flag complex $\partial_Δ\mathbf X$, called the \emph{simplicial boundary}, which is a natural setting for studying non-hyperbolic behavior of $\mathbf X$. We compare $\partial_Δ\mathbf X$ to the Roller, visual, and Tits boundaries of $\mathbf X$ and give conditions under which the natural CAT(1) metric on $\partial_Δ\mathbf X$ makes it (quasi)isometric to the Tits boundary. $\partial_Δ\mathbf X$ allows us to interpolate between studying geodesic rays in $\mathbf X$ and the geometry of its \emph{contact graph} $Γ\mathbf X$, which is known to be quasi-isometric to a tree, and we characterize essential cube complexes for which the contact graph is bounded. Using related techniques, we study divergence of combinatorial geodesics in $\mathbf X$ using $\partial_Δ\mathbf X$. Finally, we rephrase the rank-rigidity theorem of Caprace-Sageev in terms of group actions on $Γ\mathbf X$ and $\partial_Δ\mathbf X$ and state characterizations of cubulated groups with linear divergence in terms of $Γ\mathbf X$ and $\partial_Δ\mathbf X$.
We correct Theorem 3.10 of "The simplicial boundary of a CAT(0) cube complex" in the infinite-dimensional case. No correction is needed in the finite-dimensional case.
Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set, together with a pairwise-disjoint collection of subcomplexes corresponding, in the appropriate sense, to elements of P. As a special case of this result is a new proof, in the cubical case, of a Theorem of Hruska--Kleiner regarding Tits boundaries of relatively hyperbolic CAT(0) spaces. Second, we relate the existence of cut-points in asymptotic cones of a cub…
We provide a necessary and sufficient condition on a finite flag simplicial complex, L, for which there exists a unique CAT(0) cube complex whose vertex links are all isomorphic to L. We then find new examples of such CAT(0) cube complexes and prove that their automorphism groups are virtually simple. The latter uses a result, which we prove in the appendix, about the simplicity of certain subgroups of the automorphism group of a rank-one CAT(0) cube complex. This result generalizes previous results by Tits and by Haglund and Paulin.
We prove that the simplicial boundary of a CAT(0) cube complex admitting a proper, cocompact action by a virtually $\integers^n$ group is isomorphic to the hyperoctahedral triangulation of $S^{n-1}$, providing a class of groups $G$ for which the simplicial boundary of a $G$-cocompact cube complex depends only on $G$. We also use this result to show that the cocompactly cubulated crystallographic groups in dimension $n$ are precisely those that are \emph{hyperoctahedral}. We apply this result to answer a question of Wise on cocompactly cubulating virtually free abelian groups.
In [4], Dunwoody defined resolutions for finitely presented group actions on simplicial trees, that is, an action of the group on a tree with smaller edge and vertex stabilizers. He, moreover, proved that the size of the resolution is bounded by a constant depending only on the group. Extending Dunwoody's definition of patterns we construct resolutions for group actions on a general finite dimensional CAT(0) cube complex. In dimension two, we bound the number of hyperplanes of this resolution. We apply this result for surfaces and 3-manifolds to bound collections of codimension-1 submanifolds.
We study the collapsibility of finite simplicial complexes of dimension 3 endowed with a CAT(0) metric. Our main result states that, under an additional hypothesis, finite simplicial 3-complexes endowed with a CAT(0) metric collapse to a point through CAT(0) subspaces.
We describe a higher dimensional analogue of the Stallings folding sequence for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups which act properly co-compactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers.
In this short note, we show that a group acting geometrically on a CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose virtually as a free product of free abelian groups and surface groups.
We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rank one isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube complexes. We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits Alternative, an existence result for regular elements in (possibly non-uniform) lattices acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology.
In this article, we associate to isometries of CAT(0) cube complexes specific subspaces, referred to as \emph{median sets}, which play a similar role as minimising sets of semisimple isometries in CAT(0) spaces. Various applications are deduced, including a cubulation of centralisers, a splitting theorem, a proof that Dehn twists in mapping class groups must be elliptic for every action on a CAT(0) cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on CAT(0) cube complexes.
The notions of nonpositive curved spaces and biautomatic groups are generalizations of the geometric properties of hyperbolic spaces and computational properties of their fundamental groups. Given the mutual origins of these conditions, one might conjecture that groups acting on nonpositive curved spaces are biautomatic. While the conventional wisdom is that counterexamples should exist, some groups acting on nonpositively curved spaces have been shown to be biautomatic. This article adds to the list of positive examples by proving that groups acting on CAT(0) simplicial 3-complexes are biauto…
Rips and Sela introduced the notion of globally stable cylinders and asked if all Gromov hyperbolic groups admit such. We prove that hyperbolic cubulated groups admit globally stable cylinders.
In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group $F$, the lamplighter group $F \wr \mathbb{F}_2$ over a free group does not act properly on a finite-dimensional CAT(0) cube complex (although it acts properly on a infinite-dimensional CAT(0) cube complex). We also deduce from this general criterion that, roughly speaking, given a group $G$ acting on a CAT(0) cube complex of finite dimension and an infinite torsion subgroup $L \le…
This paper is a survey dedicated to the following question: given a group acting on some CAT(0) cube complex, how to exploit this action to determine whether or not the group is Gromov / relatively / acylindrically hyperbolic? As much as possible, the different criteria we mention are illustrated by applications. We also propose a model for universal acylindrical actions of cubulable groups and give a few applications to Morse, stable and hyperbolically embedded subgroups.
In this paper we prove that the asymptotic dimension of a finite-dimensional CAT(0) cube complex is bounded above by the dimension. To achieve this we prove a controlled colouring theorem for the complex. We also show that every CAT(0) cube complex is a contractive retraction of an infinite dimensional cube. As an example of the dimension theorem we obtain bounds on the asymptotic dimension of small cancellation groups.
We study the acylindrical hyperbolicity of groups acting by isometries on CAT(0) cube complexes, and obtain simple criteria formulated in terms of stabilisers for the action. Namely, we show that a group acting essentially and non-elementarily on a finite dimensional irreducible CAT(0) cube complex is acylindrically hyperbolic if there exist two hyperplanes whose stabilisers intersect along a finite subgroup. We also give further conditions on the geometry of the complex so that the result holds if we only require the existence of a single pair of points whose stabilisers intersect along a fin…
The question which motivates the article is the following: given a group acting on a CAT(0) cube complex, how can we prove that it is acylindrically hyperbolic? Keeping this goal in mind, we show a weak acylindricity of the action on the contact graph associated to a CAT(0) cube complex, and we prove a characterisation of WPD contracting isometries of CAT(0) cube complexes. As a first consequence, we find alternative arguments to show several criteria which were proved by Indira Chatterji and Alexandre Martin. Next, we show that if a group acts essentially and acylindrically on the hyperplanes…
These notes grew out of two lectures I have given on CAT(0) cube complexes. I've tried to keep the material elementary and self-contained in order to keep the material easily accessible and to provide an elementary introduction on the topic for advanced bachelor students or early master students with little to no previous knowledge on geometric group theory or CAT(0) geometry.
We introduce a $\mathbb{Z}$-valued cross ratio on Roller boundaries of ${\rm CAT(0)}$ cube complexes. We motivate its relevance by showing that every cross-ratio preserving bijection of Roller boundaries uniquely extends to a cubical isomorphism. Our results are strikingly general and even apply to infinite dimensional, locally infinite cube complexes with trivial automorphism group.
We develop the foundations of the theory of relatively geometric actions of relatively hyperbolic groups on CAT(0) cube complexes, a notion introduced in our previous work [5]. In the relatively geometric setting we prove: full relatively quasi-convex subgroups are convex compact; an analog of Agol's Theorem; and a version of Haglund--Wise's Canonical Completion and Retraction.
Given a CAT(0) cube complex X, we show that if Aut(X) $\neq$ Isom(X) then there exists a full subcomplex of X which decomposes as a product with $\mathbb{R}^n$. As applications, we prove that if X is $δ$-hyperbolic, cocompact and 1-ended, then Aut(X) $=$ Isom(X) unless X is quasi-isometric to $\mathbb{H}^2$, and extend the rank-rigidity result of Caprace-Sageev to any lattice $ Γ\leq$ Isom(X).
In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fund…
We show that every $n$-quasiflat in a $n$-dimensional $CAT(0)$ cube complex is at finite Hausdorff distance from a finite union of $n$-dimensional orthants. Then we introduce a class of cube complexes, called {\em weakly special} cube complexes and show that quasi-isometries between their universal coverings preserve top dimensional flats. We use this to establish several quasi-isometry invariants for right-angled Artin groups. Some of our arguments also extend to $CAT(0)$ spaces of finite geometric dimension. In particular, we give a short proof of the fact that a top dimensional quasiflat in…
In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups, which is a particular case of a result due to Behrstock, Caprace and Hagen. A second application gives the acylindrical hyperbolicity of $C'(1/4)-T(4)$ small cancellation quotients of free products.
The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) cube complex, and we show that contracting isometries are strongly related to isolated points at infinity, when the complex is locally finite. This boundary turns out to appear naturally in the context of Guba and Sapir's diagram groups, and we apply our main criterion to determine precisely when an element of a diagram group induces a contracting isometry on the associated …
For each d we construct CAT(0) cube complexes on which Cremona groups rank d act by isometries. From these actions we deduce new and old group theoretical and dynamical results about Cremona groups. In particular, we study the dynamical behaviour of the irreducible components of exceptional loci, we prove regularization theorems, we find new constraints on the degree growth for non-regularizable birational transformations, and we show that the centralizer of certain birational transformations is small.
We discuss a problem posed by Gersten: Is every automatic group which does not contain Z+Z subgroup, hyperbolic? To study this question, we define the notion of "n-tracks of length n", which is a structure like Z+Z, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, we show that if a group acts effectively, cellularly, properly discontinuously and cocompactly on a CAT(0) cube complex and its quotient is "weakly special", then the above question is answered affirmatively.
