586 research outputs found
Exceptional hyperbolic 3-manifolds
We correct and complete a conjecture of D. Gabai, R. Meyerhoff and N.
Thurston on the classification and properties of thin tubed closed hyperbolic
3-manifolds. We additionally show that if N is a closed hyperbolic 3-manifold,
then either N=Vol3 or N contains a closed geodesic that is the core of an
embedded tube of radius log(3)/2.Comment: 22 pages, 2 figure
Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling
An L-space is a rational homology 3-sphere with minimal Heegaard Floer
homology. We give the first examples of hyperbolic L-spaces with no symmetries.
In particular, unlike all previously known L-spaces, these manifolds are not
double branched covers of links in S^3. We prove the existence of infinitely
many such examples (in several distinct families) using a mix of hyperbolic
geometry, Floer theory, and verified computer calculations. Of independent
interest is our technique for using interval arithmetic to certify symmetry
groups and non-existence of isometries of cusped hyperbolic 3-manifolds. In the
process, we give examples of 1-cusped hyperbolic 3-manifolds of Heegaard genus
3 with two distinct lens space fillings. These are the first examples where
multiple Dehn fillings drop the Heegaard genus by more than one, which answers
a question of Gordon.Comment: 19 pages, 2 figures. v2: minor changes to intro. v3: accepted
version, to appear in Math. Res. Letter
All the shapes of spaces: a census of small 3-manifolds
In this work we present a complete (no misses, no duplicates) census for
closed, connected, orientable and prime 3-manifolds induced by plane graphs
with a bipartition of its edge set (blinks) up to edges. Blinks form a
universal encoding for such manifolds. In fact, each such a manifold is a
subtle class of blinks, \cite{lins2013B}. Blinks are in 1-1 correpondence with
{\em blackboard framed links}, \cite {kauffman1991knots, kauffman1994tlr} We
hope that this census becomes as useful for the study of concrete examples of
3-manifolds as the tables of knots are in the study of knots and links.Comment: 31 pages, 17 figures, 38 references. In this version we introduce
some new material concerning composite manifold
Spectrally Similar Incommensurable 3-Manifolds
Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3–manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for every n ≫ 0, we construct a pair of incommensurable hyperbolic 3–manifolds Nn and Nµn whose volume is approximately n and whose length spectra agree up to length n.
Both Nn and Nµn are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that these manifolds have a very thick collar about an essential surface. The two gluing maps differ by a hyper-elliptic involution along this surface. Our proof also involves a new commensurability criterion based on pairs of pants
Towards Kaluza-Klein Dark Matter on Nilmanifolds
We present a first study of the field spectrum on a class of
negatively-curved compact spaces: nilmanifolds or twisted tori. This is a case
where analytical results can be obtained, allowing to check numerical methods.
We focus on the Kaluza-Klein expansion of a scalar field. The results are then
applied to a toy model where a natural Dark Matter candidate arises as a stable
massive state of the bulk scalar.Comment: 32 pages, 8 figures; v2: few additions, published versio
Spectrally Similar Incommensurable 3-Manifolds
Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3–manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for every n ≫ 0, we construct a pair of incommensurable hyperbolic 3–manifolds Nn and Nµn whose volume is approximately n and whose length spectra agree up to length n.
Both Nn and Nµn are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that these manifolds have a very thick collar about an essential surface. The two gluing maps differ by a hyper-elliptic involution along this surface. Our proof also involves a new commensurability criterion based on pairs of pants
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