Simplices of maximal volume in hyperbolic space, Gromov's norm, and Gromov's proof of Mostow's rigidity theorem (following Thurston)

1980 ◽  
pp. 109-124 ◽  
Author(s):  
Hans J. Munkholm
Keyword(s):  
Hyperbolic Space ◽  
Rigidity Theorem ◽  
Maximal Volume

scholarly journals Rigidity at infinity for lattices in rank-one Lie groups

2018 ◽  
Vol 12 (01) ◽  
pp. 113-130 ◽  
Author(s):  
Alessio Savini
Keyword(s):  
Lie Groups ◽  
Representation Formula ◽  
Rigidity Theorem ◽  
Uniform Lattice ◽  
Maximal Volume ◽  
Reducible Representation ◽  
Totally Geodesic ◽  
Rank One ◽  
Text Component ◽  
Standard Lattice

Let [Formula: see text] be a non-uniform lattice in [Formula: see text] without torsion and with [Formula: see text]. By following the approach developed in [S. Francaviglia and B. Klaff, Maximal volume representations are Fuchsian, Geom. Dedicata 117 (2006) 111–124], we introduce the notion of volume for a representation [Formula: see text] where [Formula: see text]. We use this notion to generalize the Mostow–Prasad rigidity theorem. More precisely, we show that given a sequence of representations [Formula: see text] such that [Formula: see text], then there must exist a sequence of elements [Formula: see text] such that the representations [Formula: see text] converge to a reducible representation [Formula: see text] which preserves a totally geodesic copy of [Formula: see text] and whose [Formula: see text]-component is conjugated to the standard lattice embedding [Formula: see text]. Additionally, we show that the same definitions and results can be adapted when [Formula: see text] is a non-uniform lattice in [Formula: see text] without torsion and for representations [Formula: see text], still maintaining the hypothesis [Formula: see text].

scholarly journals An almost rigidity theorem and its applications to noncompact RCD(0,N) spaces with linear volume growth

2018 ◽  
Vol 22 (04) ◽  
pp. 1850076 ◽  
Author(s):  
Xian-Tao Huang
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Level Sets ◽  
Harmonic Functions ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Rigidity Theorem ◽  
Existence Problem ◽  
Maximal Volume ◽  
Almost Rigidity

The main results of this paper consist of two parts. First, we obtain an almost rigidity theorem which roughly says that on an [Formula: see text] space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Second, we apply this almost rigidity theorem to study noncompact [Formula: see text] spaces with linear volume growth. More precisely, we obtain the sublinear growth of diameter of geodesic spheres, and study the non-existence problem of nonconstant harmonic functions with polynomial growth on such [Formula: see text] spaces.

scholarly journals An extension of Ratner's rigidity theorem to n-dimensional hyperbolic space

1987 ◽  
Vol 7 (1) ◽  
pp. 73-92 ◽  
Author(s):  
Livio Flaminio
Keyword(s):  
Hyperbolic Space ◽  
Negative Curvature ◽  
Rigidity Theorem ◽  
Compact Manifolds ◽  
Constant Negative Curvature

AbstractWe prove that the horospherical foliations of two compact manifolds of constant negative curvature are measurably isomorphic if and only if the two manifolds are isometric.

Distance Distribution between Complex Network Nodes in Hyperbolic Space

2016 ◽  
Vol 25 (3) ◽  
pp. 223-236 ◽  
Author(s):  
Gregorio Alanis-Lobato ◽  
Miguel A. Andrade-Navarro ◽  
Keyword(s):  
Complex Network ◽  
Hyperbolic Space ◽  
Distance Distribution ◽  
Network Nodes

Density of tube packings in hyperbolic space

2004 ◽  
Vol 214 (1) ◽  
pp. 127-145 ◽  
Author(s):  
Andrew Przeworski
Keyword(s):  
Hyperbolic Space

Convergence theorems for total asymptotically nonexpansive single-valued and quasi nonexpansive multi-valued mappings in hyperbolic spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Preeyalak Chuadchawna ◽  
Ali Farajzadeh ◽  
Anchalee Kaewcharoen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Hyperbolic Space ◽  
Hyperbolic Spaces ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Uniformly Convex Hyperbolic Space

Abstract In this paper, we discuss the Δ-convergence and strong convergence for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a total asymptotically nonexpansive single-valued mapping and a quasi nonexpansive multi-valued mapping in a complete uniformly convex hyperbolic space. Finally, by giving an example, we illustrate our result.

scholarly journals Profinite rigidity for twisted Alexander polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jun Ueki
Keyword(s):  
Rigidity Theorem ◽  
Homology Groups ◽  
Alexander Polynomials ◽  
Twisted Homology ◽  
Hyperbolic Volumes

AbstractWe formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a {{\mathbb{Z}}}-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

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