225 research outputs found
Long geodesics in subgraphs of the cube
A path in the hypercube is said to be a geodesic if no two of its edges
are in the same direction. Let be a subgraph of with average degree
. How long a geodesic must contain? We show that must contain a
geodesic of length . This result, which is best possible, strengthens a
theorem of Feder and Subi. It is also related to the `antipodal colourings'
conjecture of Norine.Comment: 8 page
Polyhedral computational geometry for averaging metric phylogenetic trees
This paper investigates the computational geometry relevant to calculations
of the Frechet mean and variance for probability distributions on the
phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of
probability measures on spaces of nonpositive curvature developed by Sturm. We
show that the combinatorics of geodesics with a specified fixed endpoint in
tree space are determined by the location of the varying endpoint in a certain
polyhedral subdivision of tree space. The variance function associated to a
finite subset of tree space has a fixed algebraic formula within
each cell of the corresponding subdivision, and is continuously differentiable
in the interior of each orthant of tree space. We use this subdivision to
establish two iterative methods for producing sequences that converge to the
Frechet mean: one based on Sturm's Law of Large Numbers, and another based on
descent algorithms for finding optima of smooth functions on convex polyhedra.
We present properties and biological applications of Frechet means and extend
our main results to more general globally nonpositively curved spaces composed
of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5,
added counter example for polyhedrality of vistal subdivision in general
CAT(0) cubical complexes; v1: 43 pages, 5 figure
The simplicial boundary of a CAT(0) cube complex
For a CAT(0) cube complex , we define a simplicial flag complex
, called the \emph{simplicial boundary}, which is a
natural setting for studying non-hyperbolic behavior of . We compare
to the Roller, visual, and Tits boundaries of
and give conditions under which the natural CAT(1) metric on
makes it (quasi)isometric to the Tits boundary.
allows us to interpolate between studying geodesic
rays in and the geometry of its \emph{contact graph} , 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 using
. Finally, we rephrase the rank-rigidity theorem of
Caprace-Sageev in terms of group actions on and
and state characterizations of cubulated groups with
linear divergence in terms of and .Comment: 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
Ramified rectilinear polygons: coordinatization by dendrons
Simple rectilinear polygons (i.e. rectilinear polygons without holes or
cutpoints) can be regarded as finite rectangular cell complexes coordinatized
by two finite dendrons. The intrinsic -metric is thus inherited from the
product of the two finite dendrons via an isometric embedding. The rectangular
cell complexes that share this same embedding property are called ramified
rectilinear polygons. The links of vertices in these cell complexes may be
arbitrary bipartite graphs, in contrast to simple rectilinear polygons where
the links of points are either 4-cycles or paths of length at most 3. Ramified
rectilinear polygons are particular instances of rectangular complexes obtained
from cube-free median graphs, or equivalently simply connected rectangular
complexes with triangle-free links. The underlying graphs of finite ramified
rectilinear polygons can be recognized among graphs in linear time by a
Lexicographic Breadth-First-Search. Whereas the symmetry of a simple
rectilinear polygon is very restricted (with automorphism group being a
subgroup of the dihedral group ), ramified rectilinear polygons are
universal: every finite group is the automorphism group of some ramified
rectilinear polygon.Comment: 27 pages, 6 figure
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