99 research outputs found

    Unavoidable Immersions and Intertwines of Graphs

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    The topological minor and the minor relations are well-studied binary relations on the class of graphs. A natural weakening of the topological minor relation is an immersion. An immersion of a graph H into a graph G is a map that injects the vertex set of H into the vertex set of G such that edges between vertices of H are represented by pairwise-edge-disjoint paths of G. In this dissertation, we present two results: the first giving a set of unavoidable immersions of large 3-edge-connected graphs and the second on immersion intertwines of infinite graphs. These results, along with the methods used to prove them, are analogues of results on the graph minor relation. A conjecture for the unavoidable immersions of large 3-edge-connected graphs is also stated with a partial proof

    Characterizing the Cantor bi-cube in asymptotic categories

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    We present the characterization of metric spaces that are micro-, macro- or bi-uniformly equivalent to the extended Cantor set \{\sum_{i=-n}^\infty\frac{2x_i}{3^i}:n\in\IN ,\;(x_i)_{i\in\IZ}\in\{0,1\}^\IZ\}\subset\IR, which is bi-uniformly equivalent to the Cantor bi-cube 2^{<\IZ}=\{(x_i)_{i\in\IZ}\in \{0,1\}^\IZ:\exists n\;\forall i\ge n\;x_i=0\} endowed with the metric d((x_i),(y_i))=\max_{i\in\IZ}2^i|x_i-y_i|. Those characterizations imply that any two (uncountable) proper isometrically homogeneous ultrametric spaces are coarsely (and bi-uniformly) equivalent. This implies that any two countable locally finite groups endowed with proper left-invariant metrics are coarsely equivalent. For the proof of these results we develop a technique of towers which can have an independent interest.Comment: 24 pages; the paper now contains three characterization theorems for the extended Cantor set, which resolves all open problems posed in the preceding version of the pape

    Constant mean curvature spheres in homogeneous three-manifolds

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    We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for which such spheres exist.National Science Foundation (NSF) DMS - 1309236MINECO/FEDER MTM2014-52368-P MTM2017-89677-P MTM2016-80313-PPrograma de Apoyo a la Investigacion, Fundacion Seneca-Agencia de Ciencia y Tecnologia Region de Murcia 19461/PI/14Regional J. Andalucia P18-FR-404

    Rips construction without unique product

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    Given a finitely presented group Q,Q, we produce a short exact sequence 1→Nâ†ȘG↠Q→11\to N \hookrightarrow G \twoheadrightarrow Q \to 1 such that GG is a torsion-free Gromov hyperbolic group without the unique product property and NN is without the unique product property and has Kazhdan's Property (T). Varying Q,Q, we show a wide diversity of concrete examples of Gromov hyperbolic groups without the unique product property. As an application, we obtain Tarski monster groups without the unique product property.Comment: 22 page
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