8,204 research outputs found

### 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

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