Automorphism groups of three-dimensional hyperbolic manifolds

1992 ◽  
pp. 107-119 ◽  
Author(s):  
A. Mednykh
Keyword(s):  
Automorphism Groups ◽  
Three Dimensional ◽  
Hyperbolic Manifolds

scholarly journals Algebraically hyperbolic manifolds have finite automorphism groups

2019 ◽  
Vol 22 (02) ◽  
pp. 1950003
Author(s):  
Fedor A. Bogomolov ◽  
Ljudmila Kamenova ◽  
Misha Verbitsky
Keyword(s):  
Positive Constant ◽  
Classical Result ◽  
Automorphism Groups ◽  
Hyperbolic Manifolds ◽  
Projective Manifold ◽  
Kobayashi Hyperbolicity ◽  
Genus Formula ◽  
Projective Manifolds

A projective manifold [Formula: see text] is algebraically hyperbolic if there exists a positive constant [Formula: see text] such that the degree of any curve of genus [Formula: see text] on [Formula: see text] is bounded from above by [Formula: see text]. A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here, we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.

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 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.

Automorphism groups of 3-dimensional complex tori

1999 ◽  
Vol 1999 (508) ◽  
pp. 99-125 ◽  
Author(s):  
Ch Birkenhake ◽  
V González ◽  
H Lange
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Three Dimensional ◽  
Maximal Order ◽  
3 Dimensional ◽  
Complex Tori

Abstract We compute all finite automorphism groups of three-dimensional complex tori which are maximal in the isogeny class. The maximal order of such an automorphism group is 1296.

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