# Embedding arithmetic hyperbolic manifolds

```@article{Kolpakov2017EmbeddingAH,
title={Embedding arithmetic hyperbolic manifolds},
author={Alexander Kolpakov and Alan W. Reid and Leone Slavich},
journal={Mathematical Research Letters},
year={2017},
volume={25},
pages={1305-1328}
}```
• Published 30 March 2017
• Mathematics
• Mathematical Research Letters
We prove that any arithmetic hyperbolic n-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic (n + 1)-manifold or its universal mod 2 abelian cover can.
18 Citations
Infinitely many arithmetic hyperbolic rational homology 3-spheres that bound geometrically
• Mathematics
• 2022
In this note we provide the first examples of (arithmetic) hyperbolic 3–manifolds that are rational homology spheres and bound geometrically either compact or cusped hyperbolic 4–manifolds.
EMBEDDING CLOSED HYPERBOLIC 3-MANIFOLDS IN SMALL VOLUME HYPERBOLIC 4-MANIFOLDS
In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.
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In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.
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Algebraic & Geometric Topology
• 2021
In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.
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• Mathematics
Transactions of the American Mathematical Society
• 2019
We show that the number of isometry classes of cusped hyperbolic 3 3 -manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and
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• 2020
This paper shows that many hyperbolic manifolds obtained by glueing arithmetic pieces embed into higher-dimensional hyperbolic manifolds as codimension-one totally geodesic submanifolds. As a
Many cusped hyperbolic 3-manifolds do not bound geometrically
• Mathematics
Proceedings of the American Mathematical Society
• 2020
In this note we show that there exist cusped hyperbolic 3 3 -manifolds that embed geodesically but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for
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• Mathematics
• 2019
We give a new proof of a result of Sullivan establishing that all finite volume hyperbolic n-manifolds have a finite cover admitting a spin structure. In addition, in all dimensions ≥ 5 we give the
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Characters in Low-Dimensional Topology
• 2020
We show the existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants, addressing a conjecture of Claude LeBrun. This is achieved by showing, using results in geometric and
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• Mathematics
• 2020
We study the geography of non-compact, oriented, hyperbolic 4-manifolds of finite volume and show, in particular, that every integer is the signature of such a manifold. The main ingredients are a

## References

SHOWING 1-10 OF 91 REFERENCES
Hyperplane sections in arithmetic hyperbolic manifolds
• Mathematics
J. Lond. Math. Soc.
• 2011
It is proved that the homology groups of immersed totally geodesic hypersurfaces of compact arithmetic hyperbolic manifolds virtually inject in the homological group of the homologists of the manifolds.
Systoles of hyperbolic 4-manifolds
We prove that for any \e>0, there exists a closed hyperbolic 4-manifold with a closed geodesic of length < \e.
On the geometric boundaries of hyperbolic 4{manifolds
• Mathematics
• 2000
We provide, for hyperbolic and flat 3{manifolds, obstructions to bounding hyperbolic 4{manifolds, thus resolving in the negative a question of Farrell and Zdravkovska.
Hyperbolic four-manifolds
This is a short survey on finite-volume hyperbolic four-manifolds. We describe some general theorems and focus on the concrete examples that we found in the literature. The paper contains no new
Conformal Geometry of Discrete Groups and Manifolds
Geometric structures discontinuous groups of homeomorphisms basics of hyperbolic manifolds geometrical finiteness Kleinian manifolds uniformization theory of deformations.
Totally Geodesic Spectra of Arithmetic Hyperbolic Spaces
In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic \$n\$-orbifold, \$n\ge 4\$. Many of the results are more general and apply
Controlling manifold covers of orbifolds
In this article we prove a generalization of Selberg's lemma on the existence of torsion free, finite index subgroups of arithmetic groups. Some of the geometric applications are the resolution a
Nonarithmetic hyperbolic manifolds and trace rings
We give a sufficient condition on the hyperplanes used in the inbreeding construction of Belolipetsky-Thomson to obtain nonarithmetic manifolds. We construct explicitly infinitely many examples of
A geometrically bounding hyperbolic link complement
A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume
Non-Arithmetic Hyperbolic Reflection Groups in Higher Dimensions
We construct examples of non-arithmetic (non-cocompact) cofinite discrete reflection groups in n-dimensional Lobachevsky spaces Ln for n ≤ 18, n = 13, 15, 16, 17.