scholarly journals Truncated tetrahedra and their reflection groups

1998 ◽  
Vol 64 (1) ◽  
pp. 54-72 ◽  
Author(s):  
T. H. Marshall
Keyword(s):  
Hyperbolic Space ◽  
Hyperbolic Manifolds ◽  
Dihedral Angles ◽  
Manifolds With Boundary ◽  
Reflection Groups ◽  
Geodesic Boundary ◽  
Totally Geodesic ◽  
Generating Sets ◽  
Free Subgroups ◽  
Torsion Free

AbstractWe outline the classification, up to isometry, of all tetrahedra in hyperbolic space with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncations are all π/2, and those remaining are all submultiples of π. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups.For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary. In particular, for each g ≥ 2, we find a sequence of hyperbolic manifolds with totally geodesic boundary of genus g, which we conjecture to be of least volume among such manifolds.

scholarly journals Hyperbolic groups with boundary an n-dimensional Sierpinski space

2019 ◽  
Vol 11 (01) ◽  
pp. 233-247
Author(s):  
Jean-François Lafont ◽  
Bena Tshishiku
Keyword(s):  
Fundamental Group ◽  
Negative Curvature ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Manifolds With Boundary ◽  
Geodesic Boundary ◽  
Totally Geodesic ◽  
Visual Boundary ◽  
Aspherical Manifolds ◽  
Torsion Free

For [Formula: see text], we show that if [Formula: see text] is a torsion-free hyperbolic group whose visual boundary [Formula: see text] is an [Formula: see text]-dimensional Sierpinski space, then [Formula: see text] for some aspherical [Formula: see text]-manifold [Formula: see text] with non-empty boundary. Concerning the converse, we construct, for each [Formula: see text], examples of aspherical manifolds with boundary, whose fundamental group [Formula: see text] is hyperbolic, but with visual boundary [Formula: see text] not homeomorphic to [Formula: see text]. Our examples even support (metric) negative curvature, and have totally geodesic boundary.

Chains that realize the Gromov invariant of hyperbolic manifolds

1997 ◽  
Vol 17 (3) ◽  
pp. 643-648 ◽  
Author(s):  
DOUGLAS JUNGREIS
Keyword(s):  
Hyperbolic Manifold ◽  
Homology Class ◽  
Hyperbolic Manifolds ◽  
Uniform Measure ◽  
Geodesic Boundary ◽  
Maximal Volume ◽  
Totally Geodesic ◽  
Gromov Norm

For any closed hyperbolic manifold of dimension $n \geq 3$, suppose a sequence of $n$-cycles representing the fundamental homology class have norms converging to the Gromov invariant. We show that this sequence must converge to the uniform measure on the space of maximal-volume ideal simplices. As a corollary, we show that for a hyperbolic $n$-manifold $L$ ($n \geq 3$) with totally-geodesic boundary, the Gromov norm of ($L,\partial L$) is strictly greater than the volume of $L$ divided by the maximal volume of an ideal $n$-simplex.

scholarly journals ISOMETRIES AND DISCRETE ISOMETRY SUBGROUPS OF HYPERBOLIC SPACES

2009 ◽  
Vol 51 (1) ◽  
pp. 31-38
Author(s):  
XI FU ◽  
XIANTAO WANG
Keyword(s):  
Hyperbolic Space ◽  
Isometry Group ◽  
Quotient Space ◽  
Geometric Approach ◽  
Hyperbolic Spaces ◽  
Open Neighbourhood ◽  
Discrete Torsion ◽  
Geometric Objects ◽  
Free Subgroups ◽  
Torsion Free

AbstractLet n be the n-dimensional hyperbolic space with n ≥ 2. Suppose that G is a discrete, sense-preserving subgroup of Isomn, the isometry group of n. Let p be the projection map from n to the quotient space M = n/G. The first goal of this paper is to prove that for any a ∈ ∂n (the sphere at infinity of n), there exists an open neighbourhood U of a in n ∪ ∂ n such that p is an isometry on U ∩ n if and only if a ∈ oΩ(G) (the domain of proper discontinuity of G). This is a generalization of the main result discussed in the work by Y. D. Kim (A theorem on discrete, torsion free subgroups of Isomn, Geometriae Dedicata109 (2004), 51–57). The second goal is to obtain a new characterization for the elements of Isomn by using a class of hyperbolic geometric objects: hyperbolic isosceles right triangles. The proof is based on a geometric approach.

scholarly journals An Obata-type theorem on compact Einstein manifolds with boundary

2021 ◽  
Author(s):  
Kazuo Akutagawa
Keyword(s):  
Boundary Condition ◽  
Smooth Boundary ◽  
Type Theorem ◽  
Type Inequality ◽  
Einstein Metric ◽  
Manifolds With Boundary ◽  
Einstein Manifolds ◽  
Conformal Class ◽  
Totally Geodesic ◽  
Yamabe Constant

AbstractWe show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

scholarly journals The volume of a compact hyperbolic antiprism

2018 ◽  
Vol 27 (13) ◽  
pp. 1842010
Author(s):  
Nikolay Abrosimov ◽  
Bao Vuong
Keyword(s):  
Symmetry Group ◽  
Hyperbolic Space ◽  
Convex Polyhedron ◽  
Rotational Symmetry ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dihedral Angles ◽  
Integral Formulas ◽  
Necessary And Sufficient

We consider a compact hyperbolic antiprism. It is a convex polyhedron with [Formula: see text] vertices in the hyperbolic space [Formula: see text]. This polyhedron has a symmetry group [Formula: see text] generated by a mirror-rotational symmetry of order [Formula: see text], i.e. rotation to the angle [Formula: see text] followed by a reflection. We establish necessary and sufficient conditions for the existence of such polyhedra in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formulas expressing the volume of a hyperbolic antiprism in terms of the edge lengths.

scholarly journals The simplicial volume of hyperbolic manifolds with geodesic boundary

2010 ◽  
Vol 10 (2) ◽  
pp. 979-1001 ◽  
Author(s):  
Roberto Frigerio ◽  
Cristina Pagliantini
Keyword(s):  
Hyperbolic Manifolds ◽  
Geodesic Boundary ◽  
Simplicial Volume

scholarly journals Hyperbolic Groups are Hyperhopfian

2000 ◽  
Vol 68 (1) ◽  
pp. 126-130 ◽  
Author(s):  
Robert J. Daverman
Keyword(s):  
Abelian Subgroup ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Torsion Free

AbstractThe main result indicates that every finitely generated, residually finite, torsion-free, cohopfian group having on free Abelian subgroup of rank two is hyperhopfian. The argument relies on earlier work and ideas of Hirshon. As a corollary, fundamental groups of all closed hyperbolic manifolds are hyperhopfian.

scholarly journals Volumes of hyperbolic manifolds with geodesic boundary

Topology ◽  
1994 ◽  
Vol 33 (4) ◽  
pp. 613-629 ◽  
Author(s):  
Yosuke Miyamoto
Keyword(s):  
Hyperbolic Manifolds ◽  
Geodesic Boundary
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