On Totally Geodesic Surfaces in Symmetric Spaces of Type AI

2014 ◽  
pp. 211-227
Author(s):  
Takuya Fujimaru ◽  
Akira Kubo ◽  
Hiroshi Tamaru
Keyword(s):  
Symmetric Spaces ◽  
Totally Geodesic ◽  
Geodesic Surfaces

scholarly journals Brauer equivalent number fields and the geometry of quaternionic Shimura varieties

2018 ◽  
Vol 70 (2) ◽  
pp. 675-687
Author(s):  
Benjamin Linowitz
Keyword(s):  
Symmetric Spaces ◽  
Number Fields ◽  
Shimura Varieties ◽  
Brauer Groups ◽  
Locally Symmetric Spaces ◽  
Totally Geodesic ◽  
Equivalent Number ◽  
Geodesic Surfaces

Abstract Two number fields are said to be Brauer equivalent if there is an isomorphism between their Brauer groups that commutes with restriction. In this paper, we prove a variety of number theoretic results about Brauer equivalent number fields (for example, they must have the same signature). These results are then applied to the geometry of certain arithmetic locally symmetric spaces. As an example, we construct incommensurable arithmetic locally symmetric spaces containing exactly the same set of proper immersed totally 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
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