1,618 research outputs found
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
Hyperbolic four-manifolds, colourings and mutations
We develop a way of seeing a complete orientable hyperbolic -manifold
as an orbifold cover of a Coxeter polytope that has a facet colouring. We also develop a way of finding
totally geodesic sub-manifolds in , and describing
the result of mutations along . As an application of our method,
we construct an example of a complete orientable hyperbolic -manifold
with a single non-toric cusp and a complete orientable hyperbolic
-manifold with a single toric cusp. Both and
have twice the minimal volume among all complete orientable
hyperbolic -manifolds.Comment: 24 pages, 11 figures; to appear in Proceedings of the London
Mathematical Societ
Colouring Lines in Projective Space
Let be a vector space of dimension over a field of order . The
-Kneser graph has the -dimensional subspaces of as its vertices,
where two subspaces and are adjacent if and only if
is the zero subspace. This paper is motivated by the problem
of determining the chromatic numbers of these graphs. This problem is trivial
when (and the graphs are complete) or when (and the graphs are
empty). We establish some basic theory in the general case. Then specializing
to the case , we show that the chromatic number is when and
when . In both cases we characterise the minimal
colourings.Comment: 19 pages; to appear in J. Combinatorial Theory, Series
Compact hyperbolic manifolds without spin structures
We exhibit the first examples of compact orientable hyperbolic manifolds that
do not have any spin structure. We show that such manifolds exist in all
dimensions . The core of the argument is the construction of a
compact orientable hyperbolic -manifold that contains a surface of
genus with self intersection . The -manifold has an odd
intersection form and is hence not spin. It is built by carefully assembling
some right angled -cells along a pattern inspired by the minimum
trisection of . The manifold is also the first
example of a compact orientable hyperbolic -manifold satisfying any of these
conditions: 1) is not generated by geodesically immersed
surfaces. 2) There is a covering that is a non-trivial bundle over
a compact surface.Comment: 23 pages, 16 figure
- …