Greatly Symmetric Totally Geodesic Surfaces and Closed Hyperbolic 3-Manifolds Which Share a Fundamental Polyhedron

Topology '90 ◽  
2011 ◽  
Author(s):  
Boris N. Apanasov ◽  
Ivan S. Gutsul
Keyword(s):  
Totally Geodesic ◽  
Fundamental Polyhedron ◽  
Geodesic Surfaces

scholarly journals Totally geodesic surfaces and homology

2006 ◽  
Vol 6 (3) ◽  
pp. 1413-1428 ◽  
Author(s):  
Jason DeBlois
Keyword(s):  
Totally Geodesic ◽  
Geodesic Surfaces

scholarly journals Non-simple geodesics in hyperbolic 3-manifolds

1994 ◽  
Vol 116 (2) ◽  
pp. 339-351
Author(s):  
Kerry N. Jones ◽  
Alan W. Reid
Keyword(s):  
Closed Geodesic ◽  
Closed Geodesics ◽  
Totally Geodesic ◽  
Geodesic Surfaces ◽  
Simple Closed Geodesics

AbstractChinburg and Reid have recently constructed examples of hyperbolic 3-manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain immersed totally geodesic surfaces there are always non-simple closed geodesics. Here we construct examples of manifolds with non-simple closed geodesics and no totally geodesic surfaces.

The Generalized Cuspidal Cohomology Problem

2006 ◽  
Vol 58 (4) ◽  
pp. 673-690 ◽  
Author(s):  
Anneke Bart ◽  
Kevin P. Scannell
Keyword(s):  
Tangent Vector ◽  
Group Cohomology ◽  
Natural Generalization ◽  
Finite Range ◽  
Totally Geodesic ◽  
First Order ◽  
Link Complement ◽  
Cuspidal Cohomology ◽  
Zero Group ◽  
Geodesic Surfaces

AbstractLet Γ ⊂ SO(3, 1) be a lattice. The well known bending deformations, introduced by Thurston and Apanasov, can be used to construct non-trivial curves of representations of Γ into SO(4, 1) when Γ\ℍ3 contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in H1(Γ, ℍ41). Our main result generalizes this construction of cohomology to the context of “branched” totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in S3 which is not infinitesimally rigid in SO(4, 1). The first order deformations of this link complement are supported on a piecewise totally geodesic 2-complex.

scholarly journals Compressing totally geodesic surfaces

2002 ◽  
Vol 118 (3) ◽  
pp. 309-328 ◽  
Author(s):  
Christopher J. Leininger
Keyword(s):  
Totally Geodesic ◽  
Geodesic Surfaces

TOTALLY GEODESIC SURFACES AND QUADRATIC FORMS

2013 ◽  
Vol 22 (13) ◽  
pp. 1350072
Author(s):  
PRADTHANA JAIPONG
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
Incompressible Surface ◽  
Totally Geodesic ◽  
Geodesic Surfaces ◽  
Figure Eight Knot ◽  
Dehn Fillings

Let M be a compact, connected, irreducible, orientable 3-manifold with torus boundary. A closed, orientable, immersed, incompressible surface F in M with no incompressible annulus joining F and ∂M compresses in at most finitely many Dehn fillings M(α). It is known that there is no universal upper bound on the number of such fillings, independent of the surface, and the figure-eight knot complement is the first example of a manifold where this phenomenon occurs. In this paper, we show that the same behavior of the figure-eight knot complement is shared by other two cusped manifolds.

scholarly journals Totally geodesic surfaces in hyperbolic 3-manifolds

1991 ◽  
Vol 34 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Alan W. Reid
Keyword(s):  
Totally Geodesic ◽  
Knot Complements ◽  
Geodesic Surfaces ◽  
Figure Eight Knot ◽  
Chern Simons

In this paper we investigate totally geodesic surfaces in hyperbolic 3-manifolds. In particular we show that if M is a compact arithmetic hyperbolic 3-manifold containing an immersion of a totally geodesic surface then it contains infinitely many commensurability classes of such surfaces. In addition we show for these M that the Chern-Simons invariant is rational.We also show, that unlike the figure-eight knot complement in S3, many knot complements in S3 do not contain an immersion of a closed totally geodesic surface.

scholarly journals SMALL CURVATURE SURFACES IN HYPERBOLIC 3-MANIFOLDS

2006 ◽  
Vol 15 (03) ◽  
pp. 379-411 ◽  
Author(s):  
CHRISTOPHER J. LEININGER
Keyword(s):  
Principal Curvatures ◽  
Totally Geodesic ◽  
Small Curvature ◽  
Counter Example ◽  
Essential Surfaces ◽  
Two Component ◽  
Geodesic Surfaces

In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S3 for which the complement contains a closed embedded totally geodesic surface. In this note, we show that one can get "as close as possible" to a counter-example. Specifically, we construct a sequence of hyperbolic knots {Kn} with complements containing closed embedded essential surfaces having principal curvatures converging to zero as n tends to infinity. We also construct a family of two-component links for which the complements contain closed embedded totally geodesic surfaces of arbitrarily large genera. In addition, we prove that a closed embedded surface with sufficiently small principal curvatures is not only quasi-Fuchsian (a result of Thurston's), but it is also either acylindrical or the boundary of a twisted I-bundle.

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