99 research outputs found
Unavoidable Immersions and Intertwines of Graphs
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
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
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
Given a finitely presented group we produce a short exact sequence such that is a torsion-free
Gromov hyperbolic group without the unique product property and is without
the unique product property and has Kazhdan's Property (T). Varying 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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