3-dimensional affine space forms and hyperbolic geometry

179 Pages·2010·1.31 MB·English

by  William M. Goldman

Affinespaceformsandhyperbolicgeometry
3-dimensional affine space forms and hyperbolic
geometry
William M. Goldman
Department ofMathematics UniversityofMaryland
Conference on Geometry, Topology and
Dynamics of Character Varieties
Institute for Mathematical Sciences
National University of Singapore
university-logo
Affinespaceformsandhyperbolicgeometry
Euclidean manifolds
When can a group G act on Rn with quotient Mn = Rn/G
a (Hausdorff) manifold?
G acts by Euclidean isometries =⇒ G finite extension of a
subgroup of translations G ∩Rn ∼= Zk(Bieberbach 1912);
A Euclidean isometry is an affine transformation
~x 7−γ→ A~x +~b
A ∈ GL(n,R),~b ∈ Rn,
where the linear part L(γ) = A is orthogonal. (A ∈ O(n))
Only finitely many topological types in each dimension.
Only one commensurability class.
university-logo
Affinespaceformsandhyperbolicgeometry
Euclidean manifolds
When can a group G act on Rn with quotient Mn = Rn/G
a (Hausdorff) manifold?
G acts by Euclidean isometries =⇒ G finite extension of a
subgroup of translations G ∩Rn ∼= Zk(Bieberbach 1912);
A Euclidean isometry is an affine transformation
~x 7−γ→ A~x +~b
A ∈ GL(n,R),~b ∈ Rn,
where the linear part L(γ) = A is orthogonal. (A ∈ O(n))
Only finitely many topological types in each dimension.
Only one commensurability class.
university-logo
Affinespaceformsandhyperbolicgeometry
Euclidean manifolds
When can a group G act on Rn with quotient Mn = Rn/G
a (Hausdorff) manifold?
G acts by Euclidean isometries =⇒ G finite extension of a
subgroup of translations G ∩Rn ∼= Zk(Bieberbach 1912);
A Euclidean isometry is an affine transformation
~x 7−γ→ A~x +~b
A ∈ GL(n,R),~b ∈ Rn,
where the linear part L(γ) = A is orthogonal. (A ∈ O(n))
Only finitely many topological types in each dimension.
Only one commensurability class.
university-logo
Affinespaceformsandhyperbolicgeometry
Euclidean manifolds
When can a group G act on Rn with quotient Mn = Rn/G
a (Hausdorff) manifold?
G acts by Euclidean isometries =⇒ G finite extension of a
subgroup of translations G ∩Rn ∼= Zk(Bieberbach 1912);
A Euclidean isometry is an affine transformation
~x 7−γ→ A~x +~b
A ∈ GL(n,R),~b ∈ Rn,
where the linear part L(γ) = A is orthogonal. (A ∈ O(n))
Only finitely many topological types in each dimension.
Only one commensurability class.
university-logo
Affinespaceformsandhyperbolicgeometry
Euclidean manifolds
When can a group G act on Rn with quotient Mn = Rn/G
a (Hausdorff) manifold?
G acts by Euclidean isometries =⇒ G finite extension of a
subgroup of translations G ∩Rn ∼= Zk(Bieberbach 1912);
A Euclidean isometry is an affine transformation
~x 7−γ→ A~x +~b
A ∈ GL(n,R),~b ∈ Rn,
where the linear part L(γ) = A is orthogonal. (A ∈ O(n))
Only finitely many topological types in each dimension.
Only one commensurability class.
university-logo
Affinespaceformsandhyperbolicgeometry
Euclidean manifolds
When can a group G act on Rn with quotient Mn = Rn/G
a (Hausdorff) manifold?
G acts by Euclidean isometries =⇒ G finite extension of a
subgroup of translations G ∩Rn ∼= Zk(Bieberbach 1912);
A Euclidean isometry is an affine transformation
~x 7−γ→ A~x +~b
A ∈ GL(n,R),~b ∈ Rn,
where the linear part L(γ) = A is orthogonal. (A ∈ O(n))
Only finitely many topological types in each dimension.
Only one commensurability class.
university-logo
Affinespaceformsandhyperbolicgeometry
Euclidean manifolds
When can a group G act on Rn with quotient Mn = Rn/G
a (Hausdorff) manifold?
G acts by Euclidean isometries =⇒ G finite extension of a
subgroup of translations G ∩Rn ∼= Zk(Bieberbach 1912);
A Euclidean isometry is an affine transformation
~x 7−γ→ A~x +~b
A ∈ GL(n,R),~b ∈ Rn,
where the linear part L(γ) = A is orthogonal. (A ∈ O(n))
Only finitely many topological types in each dimension.
Only one commensurability class.
university-logo
Affinespaceformsandhyperbolicgeometry
Euclidean manifolds
When can a group G act on Rn with quotient Mn = Rn/G
a (Hausdorff) manifold?
G acts by Euclidean isometries =⇒ G finite extension of a
subgroup of translations G ∩Rn ∼= Zk(Bieberbach 1912);
A Euclidean isometry is an affine transformation
~x 7−γ→ A~x +~b
A ∈ GL(n,R),~b ∈ Rn,
where the linear part L(γ) = A is orthogonal. (A ∈ O(n))
Only finitely many topological types in each dimension.
Only one commensurability class.
university-logo
Affinespaceformsandhyperbolicgeometry
Euclidean manifolds
When can a group G act on Rn with quotient Mn = Rn/G
a (Hausdorff) manifold?
G acts by Euclidean isometries =⇒ G finite extension of a
subgroup of translations G ∩Rn ∼= Zk(Bieberbach 1912);
A Euclidean isometry is an affine transformation
~x 7−γ→ A~x +~b
A ∈ GL(n,R),~b ∈ Rn,
where the linear part L(γ) = A is orthogonal. (A ∈ O(n))
Only finitely many topological types in each dimension.
Only one commensurability class.
university-logo