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Finite Coverings in the Hyperbolic Plane

K�roly B�r�czky
2004 Discrete & Computational Geometry  
We prove that if a (th r )-convex domain in the hyperbolic plane is covered by n ≥ 2 circular discs of radius r , then the density of the covering is larger than 2π/ √ 27.  ...  Combining our result with earlier estimates yields that if at least two non-overlapping equal circular discs cover a given circular disc in a surface of constant curvature, then the density of the covering  ...  Acknowledgement I thank a referee whose comments greatly improved the paper.  ... 
doi:10.1007/s00454-004-1101-y fatcat:a7z2cik6n5fmjabaftv3vqabbm

Page 2076 of Mathematical Reviews Vol. , Issue 2000c [page]

2000 Mathematical Reviews  
The first property considered in the paper is the finite plane intersection property (FPIP).  ...  The authors prove the following theorem: If S is geometrically finite then there exists a finite covering of Ms in which every lift of S is embedded.  ... 

The noncompact hyperbolic $3$-manifold of minimal volume

Colin C. Adams
1987 Proceedings of the American Mathematical Society  
We utilize maximal cusp volumes in order to prove that the Gieseking manifold is the unique complete noncompact hyperbolic 3-manifold of minimal hyperbolic volume.  ...  Then, since the images of D under the action of P tile the plane, the images of these disks under the action of P will form a disk-packing in the plane.  ...  Let h be the Euclidean height of the boundary horosphere of H above the x-y plane. Let P be the subgroup of hyperbolic isometries in -k\ (M) which fix {oo}.  ... 
doi:10.1090/s0002-9939-1987-0894423-8 fatcat:7bhbado7yjbkniqidg6xzjaoma

The Noncompact Hyperbolic 3-Manifold of Minimal Volume

Colin C. Adams
1987 Proceedings of the American Mathematical Society  
We utilize maximal cusp volumes in order to prove that the Gieseking manifold is the unique complete noncompact hyperbolic 3-manifold of minimal hyperbolic volume.  ...  Then, since the images of D under the action of P tile the plane, the images of these disks under the action of P will form a disk-packing in the plane.  ...  Let h be the Euclidean height of the boundary horosphere of H above the x-y plane. Let P be the subgroup of hyperbolic isometries in -k\ (M) which fix {oo}.  ... 
doi:10.2307/2046691 fatcat:o7wtwjqax5entpf4wu2lc3qlsi

Page 500 of Mathematical Reviews Vol. , Issue 2000a [page]

2000 Mathematical Reviews  
Two inequivalent (27/n)-hyperbolic knots k, and kz have the same cyclic branched covering if and only if, for some (27/n)-hyperbolic 2-component link L = k, Uk2 in S 3 which is not symmetric in kK; and  ...  Ser. 2, to appear], the author shows that the graded spaces of finite-order complex- valued invariants of oriented framed knots in a solid torus and of oriented regular plane curves without direct self-tangencies  ... 

Manifolds with many hyperbolic planes [article]

Samuel Lin, Benjamin Schmidt
2017 arXiv   pre-print
We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane.  ...  Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogenous.  ...  As (H n , h n ) Riemannian covers finite volume manifolds, it remains to prove that when n 3 and 1 = b 2 + e 2a , (M n , h n,a,b ) does not Riemannian cover a finite volume manifold.  ... 
arXiv:1612.02022v2 fatcat:76po4zxs3ze4bnpzyhdsff26ey

Curvature and Computation [article]

Jon McCammond
2014 arXiv   pre-print
When undergraduates ask me what geometric group theorists study, I describe a theorem due to Gromov which relates the groups with an intrinsic geometry like that of the hyperbolic plane to those in which  ...  In short, I describe the close but surprising connection between negative curvature and efficient computation.  ...  If you have ever studied the geometry of the hyperbolic plane, you have probably learned that there are important differences between triangles in the hyperbolic plane and triangles in the euclidean plane  ... 
arXiv:1411.7306v2 fatcat:judgmtdrrzcwzhsit3spe5g6pq

Contents [chapter]

2021 Topics in Infinite Group Theory  
The Upper Half Plane as a Hyperbolic Metric Space | 262 3.3 Hyperbolic Geodesic Metric Spaces and Hyperbolic Groups | 264 3.3.1 Properly Discontinuous Actions of Groups and Quasi Isometries | 265  ...  Freiheitssatz and Finite Subgroups of One-Relator Groups | 148 1.7.2 Newman's Spelling Theorem | 153 1.7.3 Normal Closures of Single Elements in Free Groups | 155 1.7.4 The Surface Group Conjecture  ... 
doi:10.1515/9783110673371-toc fatcat:2c6s7uedmzdnrafz53b74objmq

Searching for Hyperbolicity [article]

Ruth Charney
2017 arXiv   pre-print
It is intended as a gentle introduction to geometric group theory with a focus on the notion of hyperbolicity, a theme that has inspired the field from its inception to current-day research.  ...  This is an expository paper, based on by a talk given at the AWM Research Symposium 2017.  ...  In Z, covering the torus we see infinitely many copies of the Euclidean plane (which we refer to as "flats") and emanating from each lattice point in each of these planes is a line segment covering the  ... 
arXiv:1708.01896v1 fatcat:r76qprufvveahhpe5rvgziy7yq

Orbifolds and commensurability [article]

Genevieve S. Walsh
2010 arXiv   pre-print
These are notes based on a series of talks that the author gave at the "Interactions between hyperbolic geometry and quantum groups" conference held at Columbia University in June of 2009.  ...  I am also very grateful for the hospitality of the Harvard mathematics department and for support from the NSF.  ...  Now assume that O ∼ S with a common finite index cover X, and S ∼ P with common finite index cover Y , so that X and Y both cover S.  ... 
arXiv:1003.1335v1 fatcat:gjacgik6bvcupjonuj56gochii

Asymptotic Dimension and Boundaries of Hyperbolic Spaces [article]

Thanos Gentimis
2012 arXiv   pre-print
We give an example of a visual Gromov Hyperbolic metric space X with asdimX=2 and dim(bdry X)=0  ...  We will use the standard model of the hyperbolic plane given by the interior of a disk in R 2 .  ...  Hyperbolic Plane.  ... 
arXiv:1209.2670v1 fatcat:dtaoid5srjedha7awi6e2k3osm

Complete area minimizing minimal surfaces which are not totally geodesic

Joel Hass
1984 Pacific Journal of Mathematics  
In particular, examples are described for hyperbolic space.  ...  The main theorem gives a method of constructing non-totally geodesic examples of such surfaces in certain manifolds which do not satisfy the Ricci curvature conditions.  ...  We take M λ to be the product of the hyperbolic plane with S ι and M 2 to be the product of S with the real line.  ... 
doi:10.2140/pjm.1984.111.35 fatcat:mzhfh67qyne4pd6ugkbi5kqfri

Page 6851 of Mathematical Reviews Vol. , Issue 89M [page]

1989 Mathematical Reviews  
Note the possible confusion between the terms of flock of ruled or hyperbolic quadrics and flock of a hyperbolic quadric (the former being a covering of PG(3,q) by quadrics and the latter a covering of  ...  A hyperbolic paraboloid of AG(3,q) is a hyperbolic quadric Q in PG(3, q@) less the line pair in which a tangent plane at infinity meets Q.  ... 

Hyperbolic groups with almost finitely presented subgroups [article]

Robert Kropholler, Federico Vigolo
2021 arXiv   pre-print
In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties.  ...  We give the first examples of subgroups of hyperbolic groups which are of type FP_2 but not finitely presented.  ...  We are now left in the case that these groups contain BSp1, nq for n ‰˘1. By [4] pg. 439 groups acting properly and semi-simply on CAT(0) spaces do not contain BSpm, nq for |n| ‰ |m|.  ... 
arXiv:1809.10594v2 fatcat:ismwb4shrza5bl2gsxrl2y5qaq

Page 4187 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews  
and distance, the Poincaré disc model, convex subsets of the hyperbolic plane, the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications.  ...  “Topics covered include the upper half-space model of the hyper- bolic plane, Mobius transformations, the general MObius group and the subgroup preserving path length in the upper half-space model, arc-length  ... 
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