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Finite Coverings in the Hyperbolic Plane

2004
*
Discrete & Computational Geometry
*

We prove that if a (th r )-convex domain

doi:10.1007/s00454-004-1101-y
fatcat:a7z2cik6n5fmjabaftv3vqabbm
*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. ...##
###
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

1987
*
Proceedings of the American Mathematical Society
*

We utilize maximal cusp volumes

doi:10.1090/s0002-9939-1987-0894423-8
fatcat:7bhbado7yjbkniqidg6xzjaoma
*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}. ...##
###
The Noncompact Hyperbolic 3-Manifold of Minimal Volume

1987
*
Proceedings of the American Mathematical Society
*

We utilize maximal cusp volumes

doi:10.2307/2046691
fatcat:o7wtwjqax5entpf4wu2lc3qlsi
*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}. ...##
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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]

2017
*
arXiv
*
pre-print

We construct examples of complete Riemannian manifolds having

arXiv:1612.02022v2
fatcat:76po4zxs3ze4bnpzyhdsff26ey
*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. ...##
###
Curvature and Computation
[article]

2014
*
arXiv
*
pre-print

When undergraduates ask me what geometric group theorists study, I describe a theorem due to Gromov which relates

arXiv:1411.7306v2
fatcat:judgmtdrrzcwzhsit3spe5g6pq
*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*...##
###
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 ...

##
###
Searching for Hyperbolicity
[article]

2017
*
arXiv
*
pre-print

It is intended as a gentle introduction to geometric group theory with a focus on

arXiv:1708.01896v1
fatcat:r76qprufvveahhpe5rvgziy7yq
*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*...##
###
Orbifolds and commensurability
[article]

2010
*
arXiv
*
pre-print

These are notes based on a series of talks that

arXiv:1003.1335v1
fatcat:gjacgik6bvcupjonuj56gochii
*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. ...##
###
Asymptotic Dimension and Boundaries of Hyperbolic Spaces
[article]

2012
*
arXiv
*
pre-print

We give an example of a visual Gromov

arXiv:1209.2670v1
fatcat:dtaoid5srjedha7awi6e2k3osm
*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*. ...##
###
Complete area minimizing minimal surfaces which are not totally geodesic

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. ...

##
###
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]

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|. ...

##
###
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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