340 research outputs found
A Polynomial Time Algorithm to Compute Geodesics in CAT(0) Cubical Complexes
This paper presents the first polynomial time algorithm to compute geodesics in a CAT(0) cubical complex in general dimension. The algorithm is a simple iterative method to update breakpoints of a path joining two points using Miller, Owen and Provan\u27s algorithm (Adv. in Appl. Math, 2015) as a subroutine. Our algorithm is applicable to any CAT(0) space in which geodesics between two close points can be computed, not limited to CAT(0) cubical complexes
Geodesics in CAT(0) Cubical Complexes
We describe an algorithm to compute the geodesics in an arbitrary CAT(0)
cubical complex. A key tool is a correspondence between cubical complexes of
global non-positive curvature and posets with inconsistent pairs. This
correspondence also gives an explicit realization of such a complex as the
state complex of a reconfigurable system, and a way to embed any interval in
the integer lattice cubing of its dimension.Comment: 27 pages, 7 figure
Hierarchically hyperbolic spaces I: curve complexes for cubical groups
In the context of CAT(0) cubical groups, we develop an analogue of the theory
of curve complexes and subsurface projections. The role of the subsurfaces is
played by a collection of convex subcomplexes called a \emph{factor system},
and the role of the curve graph is played by the \emph{contact graph}. There
are a number of close parallels between the contact graph and the curve graph,
including hyperbolicity, acylindricity of the action, the existence of
hierarchy paths, and a Masur--Minsky-style distance formula.
We then define a \emph{hierarchically hyperbolic space}; the class of such
spaces includes a wide class of cubical groups (including all virtually compact
special groups) as well as mapping class groups and Teichm\"{u}ller space with
any of the standard metrics. We deduce a number of results about these spaces,
all of which are new for cubical or mapping class groups, and most of which are
new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent
Lie group into a hierarchically hyperbolic space lies close to a product of
hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic
spaces; this generalizes results of Behrstock--Minsky, Eskin--Masur--Rafi,
Hamenst\"{a}dt, and Kleiner. We finally prove that each hierarchically
hyperbolic group admits an acylindrical action on a hyperbolic space. This
acylindricity result is new for cubical groups, in which case the hyperbolic
space admitting the action is the contact graph; in the case of the mapping
class group, this provides a new proof of a theorem of Bowditch.Comment: To appear in "Geometry and Topology". This version incorporates the
referee's comment
The space of ultrametric phylogenetic trees
The reliability of a phylogenetic inference method from genomic sequence data
is ensured by its statistical consistency. Bayesian inference methods produce a
sample of phylogenetic trees from the posterior distribution given sequence
data. Hence the question of statistical consistency of such methods is
equivalent to the consistency of the summary of the sample. More generally,
statistical consistency is ensured by the tree space used to analyse the
sample.
In this paper, we consider two standard parameterisations of phylogenetic
time-trees used in evolutionary models: inter-coalescent interval lengths and
absolute times of divergence events. For each of these parameterisations we
introduce a natural metric space on ultrametric phylogenetic trees. We compare
the introduced spaces with existing models of tree space and formulate several
formal requirements that a metric space on phylogenetic trees must possess in
order to be a satisfactory space for statistical analysis, and justify them. We
show that only a few known constructions of the space of phylogenetic trees
satisfy these requirements. However, our results suggest that these basic
requirements are not enough to distinguish between the two metric spaces we
introduce and that the choice between metric spaces requires additional
properties to be considered. Particularly, that the summary tree minimising the
square distance to the trees from the sample might be different for different
parameterisations. This suggests that further fundamental insight is needed
into the problem of statistical consistency of phylogenetic inference methods.Comment: Minor changes. This version has been published in JTB. 27 pages, 9
figure
Acylindrical hyperbolicity of cubical small-cancellation groups
We provide an analogue of Strebel's classification of geodesic triangles in
classical 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
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