Three-dimensional hyperbolic manifolds with general fundamental polyhedron

1991 ◽  
Vol 49 (6) ◽  
pp. 575-577 ◽  
Author(s):  
A. Yu. Vesnin
Keyword(s):  
Three Dimensional ◽  
Hyperbolic Manifolds ◽  
Fundamental Polyhedron

On the complexity of three-dimensional cusped hyperbolic manifolds

2014 ◽  
Vol 89 (3) ◽  
pp. 267-270 ◽  
Author(s):  
A. Yu. Vesnin ◽  
V. V. Tarkaev ◽  
E. A. Fominykh
Keyword(s):  
Three Dimensional ◽  
Hyperbolic Manifolds

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.

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