26,454 research outputs found
Bucolic Complexes
We introduce and investigate bucolic complexes, a common generalization of
systolic complexes and of CAT(0) cubical complexes. They are defined as simply
connected prism complexes satisfying some local combinatorial conditions. We
study various approaches to bucolic complexes: from graph-theoretic and
topological perspective, as well as from the point of view of geometric group
theory. In particular, we characterize bucolic complexes by some properties of
their 2-skeleta and 1-skeleta (that we call bucolic graphs), by which several
known results are generalized. We also show that locally-finite bucolic
complexes are contractible, and satisfy some nonpositive-curvature-like
properties.Comment: 45 pages, 4 figure
Panel collapse and its applications
We describe a procedure called panel collapse for replacing a CAT(0) cube
complex by a "lower complexity" CAT(0) cube complex
whenever contains a codimension- hyperplane that is extremal in one
of the codimension- hyperplanes containing it. Although is
not in general a subcomplex of , it is a subspace consisting of a
subcomplex together with some cubes that sit inside "diagonally". The
hyperplanes of extend to hyperplanes of . Applying this
procedure, we prove: if a group acts cocompactly on a CAT(0) cube complex
, then there is a CAT(0) cube complex so that acts
cocompactly on and for each hyperplane of , the stabiliser
in of acts on 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 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
Subquadratic-time algorithm for the diameter and all eccentricities on median graphs
On sparse graphs, Roditty and Williams [2013] proved that no
-time algorithm achieves an approximation factor smaller
than for the diameter problem unless SETH fails. In this article,
we solve an open question formulated in the literature: can we use the
structural properties of median graphs to break this global quadratic barrier?
We propose the first combinatiorial algorithm computing exactly all
eccentricities of a median graph in truly subquadratic time. Median graphs
constitute the family of graphs which is the most studied in metric graph
theory because their structure represents many other discrete and geometric
concepts, such as CAT(0) cube complexes. Our result generalizes a recent one,
stating that there is a linear-time algorithm for all eccentricities in median
graphs with bounded dimension , i.e. the dimension of the largest induced
hypercube. This prerequisite on is not necessarily anymore to determine all
eccentricities in subquadratic time. The execution time of our algorithm is
.
We provide also some satellite outcomes related to this general result. In
particular, restricted to simplex graphs, this algorithm enumerates all
eccentricities with a quasilinear running time. Moreover, an algorithm is
proposed to compute exactly all reach centralities in time
.Comment: 43 pages, extended abstract in STACS 202
Groups acting on quasi-median graphs. An introduction
Quasi-median graphs have been introduced by Mulder in 1980 as a
generalisation of median graphs, known in geometric group theory to naturally
coincide with the class of CAT(0) cube complexes. In his PhD thesis, the author
showed that quasi-median graphs may be useful to study groups as well. In the
present paper, we propose a gentle introduction to the theory of groups acting
on quasi-median graphs.Comment: 16 pages. Comments are welcom
On embeddings of CAT(0) cube complexes into products of trees
We prove that the contact graph of a 2-dimensional CAT(0) cube complex of maximum degree can be coloured with at most
colours, for a fixed constant . This implies
that (and the associated median graph) isometrically embeds in the
Cartesian product of at most trees, and that the event
structure whose domain is admits a nice labeling with
labels. On the other hand, we present an example of a
5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes
which cannot be embedded into a Cartesian product of a finite number of trees.
This answers in the negative a question raised independently by F. Haglund, G.
Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the
computation of the bounds in Theorem 1. Some figures repaire
- …