On the Fundamental Group of Compact Manifolds of Non-Positive Curvature

1971 ◽  
Vol 93 (3) ◽  
pp. 579 ◽  
Author(s):  
Shing Tung Yau
Keyword(s):  
Fundamental Group ◽  
Positive Curvature ◽  
Compact Manifolds

A note on Reidemeister torsion and pleated surfaces

2014 ◽  
Vol 23 (03) ◽  
pp. 1450015 ◽  
Author(s):  
Yasar Sozen
Keyword(s):  
Fundamental Group ◽  
Explicit Formula ◽  
Symplectic Form ◽  
Chain Complex ◽  
Hyperbolic Surface ◽  
Compact Manifolds ◽  
Complex Numbers ◽  
Manifolds With Boundary ◽  
Reidemeister Torsion ◽  
Cup Product

This paper uses the notion of ℂ-symplectic chain complex and proves an explicit formula for the Reidemeister torsion of an arbitrary ℂ-symplectic chain complex in terms of intersection forms of the homologies. In applications, the formula is applied to closed manifolds and also compact manifolds with boundary by using the homologies with coefficients in complex numbers field. Moreover, an explicit formula for the Reidemeister torsion of representations from the fundamental group of a closed oriented hyperbolic surface to PSL2(ℂ) is presented in terms of the cup product of twisted cohomologies, which is related with Weil–Petersson form and thus the Thurston symplectic form. The formula is also applied to pleated surfaces.

scholarly journals The fundamental group of compact manifolds without conjugate points

1986 ◽  
Vol 61 (1) ◽  
pp. 161-175 ◽  
Author(s):  
Christopher B. Croke ◽  
Viktor Schroeder
Keyword(s):  
Fundamental Group ◽  
Conjugate Points ◽  
Compact Manifolds

scholarly journals Simplicial volume of compact manifolds with amenable boundary

2014 ◽  
Vol 07 (01) ◽  
pp. 23-46 ◽  
Author(s):  
Sungwoon Kim ◽  
Thilo Kuessner
Keyword(s):  
Fundamental Group ◽  
Finite Volume ◽  
Compact Manifold ◽  
Riemannian Metric ◽  
Compact Manifolds ◽  
Simplicial Volume ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Path Component ◽  
Proportionality Principle

Let M be the interior of a connected, oriented, compact manifold V of dimension at least 2. If each path component of ∂V has amenable fundamental group, then we prove that the simplicial volume of M is equal to the relative simplicial volume of V and also to the geometric (Lipschitz) simplicial volume of any Riemannian metric on M whenever the latter is finite. As an application we establish the proportionality principle for the simplicial volume of complete, pinched negatively curved manifolds of finite volume.

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