Weak hyperbolicity of cube complexes and quasi-arboreal groups

We examine a graph $\Gamma$ encoding the intersection of hyperplane carriers in a CAT(0) cube complex $\widetilde X$. The main result is that $\Gamma$ is quasi-isometric to a tree. This implies that a group $G$ acting properly and cocompactly on $\widetilde X$ is weakly hyperbolic relative to the hyperplane stabilizers. Using disc diagram techniques and Wright's recent result on the aymptotic dimension of CAT(0) cube complexes, we give a generalization of a theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs of asymptotically finite-dimensional groups. More precisely, we prove asymptotic finite-dimensionality for finitely-generated groups acting on finite-dimensional cube complexes with 0-cube stabilizers of uniformly bounded asymptotic dimension. Finally, we apply contact graph techniques to prove a cubical version of the flat plane theorem stated in terms of complete bipartite subgraphs of $\Gamma$.Comment: Corrections in Sections 2 and 4. Simplification in Section

Cocompactly cubulated crystallographic groups

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.Comment: Several correction

On hierarchical hyperbolicity of cubical groups

Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. We give three conditions on the action, any one of which ensures that X has a factor system in the sense of [BHS14]. We also prove that one of these conditions is necessary. This combines with results of Behrstock--Hagen--Sisto to show that $G$ is a hierarchically hyperbolic group; this partially answers questions raised by those authors. Under any of these conditions, our results also affirm a conjecture of BehrstockHagen on boundaries of cube complexes, which implies that X cannot contain a convex staircase. The conditions on the action are all strictly weaker than virtual cospecialness, and we are not aware of a cocompactly cubulated group that does not satisfy at least one of the conditions.Comment: Minor changes in response to referee report. Streamlined the proof of Lemma 5.2, and added an examples of non-rotational action

Acylindrical hyperbolicity of cubical small-cancellation groups

We provide an analogue of Strebel's classification of geodesic triangles in classical $C'(\frac16)$ groups for groups given by Wise's cubical presentations satisfying sufficiently strong metric cubical small cancellation conditions. Using our classification, we prove that, except in specific degenerate cases, such groups are acylindrically hyperbolic.Comment: Added figures. Exposition improved in Section 3, correction/simplification in Section 5, background added and citations updated in Section

Panel collapse and its applications

We describe a procedure called panel collapse for replacing a CAT(0) cube complex $\Psi$ by a "lower complexity" CAT(0) cube complex $\Psi_\bullet$ whenever $\Psi$ contains a codimension-$2$ hyperplane that is extremal in one of the codimension-$1$ hyperplanes containing it. Although $\Psi_\bullet$ is not in general a subcomplex of $\Psi$, it is a subspace consisting of a subcomplex together with some cubes that sit inside $\Psi$ "diagonally". The hyperplanes of $\Psi_\bullet$ extend to hyperplanes of $\Psi$. Applying this procedure, we prove: if a group $G$ acts cocompactly on a CAT(0) cube complex $\Psi$, then there is a CAT(0) cube complex $\Omega$ so that $G$ acts cocompactly on $\Omega$ and for each hyperplane $H$ of $\Omega$, the stabiliser in $G$ of $H$ acts on $H$ essentially. Using panel collapse, we obtain a new proof of Stallings's theorem on groups with more than one end. As another illustrative example, we show that panel collapse applies to the exotic cubulations of free groups constructed by Wise. Next, we show that the CAT(0) cube complexes constructed by Cashen-Macura can be collapsed to trees while preserving all of the necessary group actions. (It also illustrates that our result applies to actions of some non-discrete groups.) We also discuss possible applications to quasi-isometric rigidity for certain classes of graphs of free groups with cyclic edge groups. Panel collapse is also used in forthcoming work of the first-named author and Wilton to study fixed-point sets of finite subgroups of $\mathrm{Out}(F_n)$ on the free splitting complex. Finally, we apply panel collapse to a conjecture of Kropholler, obtaining a short proof under a natural extra hypothesis.Comment: Revised according to referee comments. This version accepted in "Groups, Geometry, and Dynamics

Quasiflats in hierarchically hyperbolic spaces

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twist flats for mapping class groups, the maximal rank of a free abelian subgroup for right-angled Coxeter and Artin groups, and, for the Weil--Petersson metric, the rank is the integer part of half the complex dimension of Teichm\"{u}ller space. We prove that any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a mild condition satisfied by all natural examples). This resolves outstanding conjectures when applied to various examples. For mapping class group, we verify a conjecture of Farb; for Teichm\"{u}ller space we answer a question of Brock; for CAT(0) cubical groups, we handle special cases including right-angled Coxeter groups. An important ingredient in the proof is that the hull of any finite set in an HHS is quasi-isometric to a CAT(0) cube complex of dimension bounded by the rank. We deduce a number of applications. For instance, we show that any quasi-isometry between HHSs induces a quasi-isometry between certain simpler HHSs. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups. Another application is to quasi-isometric rigidity. Our tools in many cases allow one to reduce the problem of quasi-isometric rigidity for a given hierarchically hyperbolic group to a combinatorial problem. We give a new proof of quasi-isometric rigidity of mapping class groups, which, given our general quasiflats theorem, uses simpler combinatorial arguments than in previous proofs.Comment: 58 pages, 6 figures. Revised according to referee comments. This is the final pre-publication version; to appear in Duke Math. Jou

Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups

We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a finite type surface: improving the bound from exponential to at most quadratic in the complexity of the surface. We also apply the main result to various other hierarchically hyperbolic groups and spaces. We also prove a small-cancellation result namely: if $G$ is a hierarchically hyperbolic group, $H\leq G$ is a suitable hyperbolically embedded subgroup, and $N\triangleleft H$ is "sufficiently deep" in $H$, then $G/\langle\langle N\rangle\rangle$ is a relatively hierarchically hyperbolic group. This new class provides many new examples to which our asymptotic dimension bounds apply. Along the way, we prove new results about the structure of HHSs, for example: the associated hyperbolic spaces are always obtained, up to quasi-isometry, by coning off canonical coarse product regions in the original space (generalizing a relation established by Masur--Minsky between the complex of curves of a surface and Teichm\"{u}ller space).Comment: Minor revisions in Section 6. This is the version accepted for publicatio