1,236 research outputs found
New hyperbolic 4-manifolds of low volume
We prove that there are at least 2 commensurability classes of minimal-volume
hyperbolic 4-manifolds. Moreover, by applying a well-known technique due to
Gromov and Piatetski-Shapiro, we build the smallest known non-arithmetic
hyperbolic 4-manifold.Comment: 21 pages, 6 figures. Added the Coxeter diagrams of the
commensurability classes of the manifolds. New and better proof of Lemma 2.2.
Modified statements and proofs of the main theorems: now there are two
commensurabilty classes of minimal volume manifolds. Typos correcte
Geometric transition from hyperbolic to Anti-de Sitter structures in dimension four
We provide the first examples of geometric transition from hyperbolic to
Anti-de Sitter structures in dimension four, in a fashion similar to Danciger's
three-dimensional examples. The main ingredient is a deformation of hyperbolic
4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a
3-dimensional ideal cuboctahedron. We show the existence of a similar family of
collapsing Anti-de Sitter polytopes, and join the two deformations by means of
an opportune half-pipe orbifold structure. The desired examples of geometric
transition are then obtained by gluing copies of the polytope.Comment: 50 pages, 27 figures. Part 3 of the previous version has been removed
and will be part of a new preprint to appear soo
Hyperbolic Dehn filling in dimension four
We introduce and study some deformations of complete finite-volume hyperbolic
four-manifolds that may be interpreted as four-dimensional analogues of
Thurston's hyperbolic Dehn filling.
We construct in particular an analytic path of complete, finite-volume cone
four-manifolds that interpolates between two hyperbolic four-manifolds
and with the same volume . The deformation looks
like the familiar hyperbolic Dehn filling paths that occur in dimension three,
where the cone angle of a core simple closed geodesic varies monotonically from
to . Here, the singularity of is an immersed geodesic surface
whose cone angles also vary monotonically from to . When a cone angle
tends to a small core surface (a torus or Klein bottle) is drilled
producing a new cusp.
We show that various instances of hyperbolic Dehn fillings may arise,
including one case where a degeneration occurs when the cone angles tend to
, like in the famous figure-eight knot complement example.
The construction makes an essential use of a family of four-dimensional
deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm.Comment: 60 pages, 23 figures. Final versio
Counting cusped hyperbolic 3-manifolds that bound geometrically
We show that the number of isometry classes of cusped hyperbolic
-manifolds that bound geometrically grows at least super-exponentially with
their volume, both in the arithmetic and non-arithmetic settings.Comment: 17 pages, 7 figures; to appear in Transactions AM
Compact hyperbolic manifolds without spin structures
We exhibit the first examples of compact orientable hyperbolic manifolds that
do not have any spin structure. We show that such manifolds exist in all
dimensions . The core of the argument is the construction of a
compact orientable hyperbolic -manifold that contains a surface of
genus with self intersection . The -manifold has an odd
intersection form and is hence not spin. It is built by carefully assembling
some right angled -cells along a pattern inspired by the minimum
trisection of . The manifold is also the first
example of a compact orientable hyperbolic -manifold satisfying any of these
conditions: 1) is not generated by geodesically immersed
surfaces. 2) There is a covering that is a non-trivial bundle over
a compact surface.Comment: 23 pages, 16 figure
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