847 research outputs found
Grope cobordism of classical knots
We explain the notion of a grope cobordism between two knots in a 3-manifold.
Each grope cobordism has a type that can be described by a rooted unitrivalent
tree. By filtering these trees in different ways, we show how the
Goussarov-Habiro approach to finite type invariants of knots is closely related
to our notion of grope cobordism. Thus our results can be viewed as a geometric
interpretation of finite type invariants.
An interesting refinement we study are knots modulo symmetric grope cobordism
in 3-space. On one hand this theory maps onto the usual Vassiliev theory and on
the other hand it maps onto the Cochran-Orr-Teichner filtration of the knot
concordance group, via symmetric grope cobordism in 4-space. In particular, the
graded theory contains information on finite type invariants (with degree h
terms mapping to Vassiliev degree 2^h), Blanchfield forms or S-equivalence at
h=2, Casson-Gordon invariants at h=3, and for h=4 one has the new von Neumann
signatures of a knot.Comment: Final version. To appear in Topology. See
http://www.math.cornell.edu/~jconant/pagethree.html for a PDF file with
better figure qualit
Random subgroups of Thompson's group
We consider random subgroups of Thompson's group with respect to two
natural stratifications of the set of all generator subgroups. We find that
the isomorphism classes of subgroups which occur with positive density are not
the same for the two stratifications.
We give the first known examples of {\em persistent} subgroups, whose
isomorphism classes occur with positive density within the set of -generator
subgroups, for all sufficiently large . Additionally, Thompson's group
provides the first example of a group without a generic isomorphism class of
subgroup. Elements of are represented uniquely by reduced pairs of finite
rooted binary trees.
We compute the asymptotic growth rate and a generating function for the
number of reduced pairs of trees, which we show is D-finite and not algebraic.
We then use the asymptotic growth to prove our density results.Comment: 37 pages, 11 figure
Unimodular measures on the space of all Riemannian manifolds
We study unimodular measures on the space of all pointed
Riemannian -manifolds. Examples can be constructed from finite volume
manifolds, from measured foliations with Riemannian leaves, and from invariant
random subgroups of Lie groups. Unimodularity is preserved under weak* limits,
and under certain geometric constraints (e.g. bounded geometry) unimodular
measures can be used to compactify sets of finite volume manifolds. One can
then understand the geometry of manifolds with large, finite volume by
passing to unimodular limits.
We develop a structure theory for unimodular measures on ,
characterizing them via invariance under a certain geodesic flow, and showing
that they correspond to transverse measures on a foliated `desingularization'
of . We also give a geometric proof of a compactness theorem for
unimodular measures on the space of pointed manifolds with pinched negative
curvature, and characterize unimodular measures supported on hyperbolic
-manifolds with finitely generated fundamental group.Comment: 81 page
On the Approximability of the Traveling Salesman Problem with Line Neighborhoods
We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in , with , are -hardness and an -approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in is APX-hard for any . More generally, this implies that TSP with -dimensional flats does not admit a PTAS for any unless , which gives a complete classification of the approximability of these problems, as there are known PTASes for (i.e., points) and (hyperplanes). We are able to give a stronger inapproximability factor for by showing that TSP with lines does not admit a -approximation in dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an -approximation algorithm for the problem, albeit with a running time of
A Time Hierarchy Theorem for the LOCAL Model
The celebrated Time Hierarchy Theorem for Turing machines states, informally,
that more problems can be solved given more time. The extent to which a time
hierarchy-type theorem holds in the distributed LOCAL model has been open for
many years. It is consistent with previous results that all natural problems in
the LOCAL model can be classified according to a small constant number of
complexities, such as , etc.
In this paper we establish the first time hierarchy theorem for the LOCAL
model and prove that several gaps exist in the LOCAL time hierarchy.
1. We define an infinite set of simple coloring problems called Hierarchical
-Coloring}. A correctly colored graph can be confirmed by simply
checking the neighborhood of each vertex, so this problem fits into the class
of locally checkable labeling (LCL) problems. However, the complexity of the
-level Hierarchical -Coloring problem is ,
for . The upper and lower bounds hold for both general graphs
and trees, and for both randomized and deterministic algorithms.
2. Consider any LCL problem on bounded degree trees. We prove an
automatic-speedup theorem that states that any randomized -time
algorithm solving the LCL can be transformed into a deterministic -time algorithm. Together with a previous result, this establishes that on
trees, there are no natural deterministic complexities in the ranges
--- or ---.
3. We expose a gap in the randomized time hierarchy on general graphs. Any
randomized algorithm that solves an LCL problem in sublogarithmic time can be
sped up to run in time, which is the complexity of the distributed
Lovasz local lemma problem, currently known to be and
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