847 research outputs found

    Grope cobordism of classical knots

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    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 FF

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    We consider random subgroups of Thompson's group FF with respect to two natural stratifications of the set of all kk 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 kk-generator subgroups, for all sufficiently large kk. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of FF 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

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    We study unimodular measures on the space Md\mathcal M^d of all pointed Riemannian dd-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 MM with large, finite volume by passing to unimodular limits. We develop a structure theory for unimodular measures on Md\mathcal M^d, characterizing them via invariance under a certain geodesic flow, and showing that they correspond to transverse measures on a foliated `desingularization' of Md\mathcal M^d. 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 33-manifolds with finitely generated fundamental group.Comment: 81 page

    On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

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    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 Rd\mathbb{R}^d, with dβ‰₯3d\ge 3, are NP\mathrm{NP}-hardness and an O(log⁑3n)O(\log^3 n)-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in Rd\mathbb{R}^d is APX-hard for any dβ‰₯3d\ge 3. More generally, this implies that TSP with kk-dimensional flats does not admit a PTAS for any 1≀k≀dβˆ’21\le k \leq d-2 unless P=NP\mathrm{P}=\mathrm{NP}, which gives a complete classification of the approximability of these problems, as there are known PTASes for k=0k=0 (i.e., points) and k=dβˆ’1k=d-1 (hyperplanes). We are able to give a stronger inapproximability factor for d=O(log⁑n)d=O(\log n) by showing that TSP with lines does not admit a (2βˆ’Ο΅)(2-\epsilon)-approximation in dd 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 O(log⁑2n)O(\log^2 n)-approximation algorithm for the problem, albeit with a running time of nO(log⁑log⁑n)n^{O(\log\log n)}

    A Time Hierarchy Theorem for the LOCAL Model

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    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 O(1),O(logβ‘βˆ—n),O(log⁑n),2O(log⁑n)O(1),O(\log^* n), O(\log n), 2^{O(\sqrt{\log n})}, 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 2122\frac{1}{2}-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 kk-level Hierarchical 2122\frac{1}{2}-Coloring problem is Θ(n1/k)\Theta(n^{1/k}), for k∈Z+k\in\mathbb{Z}^+. 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 no(1)n^{o(1)}-time algorithm solving the LCL can be transformed into a deterministic O(log⁑n)O(\log n)-time algorithm. Together with a previous result, this establishes that on trees, there are no natural deterministic complexities in the ranges Ο‰(logβ‘βˆ—n)\omega(\log^* n)---o(log⁑n)o(\log n) or Ο‰(log⁑n)\omega(\log n)---no(1)n^{o(1)}. 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 O(TLLL)O(T_{LLL}) time, which is the complexity of the distributed Lovasz local lemma problem, currently known to be Ξ©(log⁑log⁑n)\Omega(\log\log n) and O(log⁑n)O(\log n)
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