Discrete groups in three-dimensional Lobachevsky space generated by two rotations

1989 ◽  
Vol 30 (1) ◽  
pp. 95-100 ◽  
Author(s):  
E. Ya. Klimenko
Keyword(s):  
Three Dimensional ◽  
Discrete Groups ◽  
Lobachevsky Space

scholarly journals On the classification of stably reflective hyperbolic Z[√2]-lattices of rank 4

2019 ◽  
Vol 486 (1) ◽  
pp. 7-11
Author(s):  
N. V. Bogachev
Keyword(s):  
Three Dimensional ◽  
Reflection Group ◽  
Lobachevsky Space ◽  
Fundamental Polyhedron

In this paper we prove that the fundamental polyhedron of a ℤ2-arithmetic reflection group in the three-dimensional Lobachevsky space contains an edge such that the distance between its framing faces is small enough. Using this fact we obtain a classification of stably reflective hyperbolic ℤ2-lattices of rank 4.

scholarly journals Borel and Volume Classes for Dense Representations of Discrete Groups

2021 ◽  
Author(s):  
James Farre
Keyword(s):  
Three Dimensional ◽  
Hyperbolic Manifolds ◽  
Discrete Groups ◽  
Bounded Cohomology ◽  
Hyperbolic Volume ◽  
Uniformly Separated ◽  
Point Line ◽  
Borel Class ◽  
Dense Representation ◽  
The Continuum

Abstract We show that the bounded Borel class of any dense representation $\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ is uniformly separated in semi-norm from any other representation $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$ for which there is a subgroup $H\le G$ on which $\rho $ is still dense but $\rho ^{\prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of ${\operatorname{PSL}}_2{\mathbb{R}}$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.

Taxonomic Revision of the Vachellia acuifera Species Group (Fabaceae: Mimosoideae) in the Carribean

2009 ◽  
Vol 34 (1) ◽  
pp. 84-101 ◽  
Author(s):  
H. David Clarke ◽  
David S. Seigler ◽  
John E. Ebinger
Keyword(s):  
Geographic Distribution ◽  
Three Dimensional ◽  
Taxonomic Revision ◽  
Virgin Islands ◽  
Species Group ◽  
The Other ◽  
Discrete Groups ◽  
The Bahamas ◽  
British Virgin Islands ◽  
Vegetative Characters

Twelve Caribbean species of Vachellia (formerlyAcacia subgenus Acacia) are formally monographed with taxonomic, habitat, geographic distribution descriptions, and lists of representative specimens examined. The species treated here are all part of the informal Vachellia acuifera group, which is characterized by generally enlarged stipular spines borne in clusters on the stems, 1–3 pairs of pinnae per leaf, and 20–50 stamens. Ordination analyses of vegetative characters show that these species form discrete groups in two- and three-dimensional plots. Vachellia baessleria Clarke, Seigler, & Ebinger, endemic to Guantánamo and Las Tunas, Cuba, is newly described. Vachellia acuifera is wide ranging, occurring from the Bahamas to Cuba. The other species are highly restricted in their distribution and ecological setting. Four of the species are only known from Hispaniola (V. barahonensis, V. caurina, V. cucuyo, and V. oviedoensis) and six only from Cuba (V. baessleria, V. belairioides, V. bucheri, V. daemon, V. roigii, and V. zapatensis). Another species, V. anagadensis, is only found on the island of Anegada in the British Virgin Islands.

Coherent states on horospheric three-dimensional Lobachevsky space

2016 ◽  
Vol 57 (8) ◽  
pp. 082111 ◽  
Author(s):  
Yu. Kurochkin ◽  
I. Rybak ◽  
Dz. Shoukavy
Keyword(s):  
Coherent States ◽  
Three Dimensional ◽  
Lobachevsky Space

scholarly journals Description of a free quantum-mechanical particle in the Lobachevsky space based on the integral equation

2019 ◽  
Vol 55 (3) ◽  
pp. 319-324
Author(s):  
Yu. A. Kurochkin
Keyword(s):  
Integral Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Scattering Amplitude ◽  
Separation Of Variables ◽  
Free Particle ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Quantum Mechanical ◽  
Lobachevsky Space

The quantum mechanical problem of the motion of a free particle in the three-dimensional Lobachevsky space is interpreted as space scattering. The quantum case is considered on the basis of the integral equation derived from the Schrödinger equation. The work continues the problem considered in [1] studied within the framework of classical mechanics and on the basis of solving the Schrödinger equation in quasi-Cartesian coordinates. The proposed article also uses a quasi-Cartesian coordinate system; however after the separation of variables, the integral equation is derived for the motion along the axis of symmetry horosphere axis coinciding with the z axis. The relationship between the scattering amplitude and the analytical functions is established. The iteration method and finite differences for solution of the integral equation are proposed.

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