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

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 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.

Weak stability of the geodesic flow and Preissmann's theorem

2000 ◽  
Vol 20 (4) ◽  
pp. 1231-1251
Author(s):  
RAFAEL OSWALDO RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Geodesic Flow ◽  
Abelian Subgroup ◽  
Conjugate Points ◽  
Weak Stability ◽  
Shadowing Property

Let $(M,g)$ be a compact, differentiable Riemannian manifold without conjugate points and bounded asymptote. We show that, if the geodesic flow of $(M,g)$ is either topologically stable, or satisfies the $\epsilon$-shadowing property for some appropriate $\epsilon > 0$, then every abelian subgroup of the fundamental group of $M$ is infinite cyclic. The proof is based on the existence of homoclinic geodesics in perturbations of $(M,g)$, whenever there is a subgroup of the fundamental group of $M$ isomorphic to $\mathbb{Z}\times \mathbb{Z}$.

scholarly journals A support theorem for the X-ray transform on manifolds with plane covers

2019 ◽  
Vol 169 (1) ◽  
pp. 149-158
Author(s):  
NORBERT PEYERIMHOFF ◽  
EVANGELIA SAMIOU
Keyword(s):  
Lie Groups ◽  
Conjugate Points ◽  
Compact Manifolds ◽  
Nilpotent Lie Groups ◽  
Support Theorem ◽  
X Ray ◽  
Ray Transform ◽  
3 Dimensional ◽  
Support Theorems ◽  
Simply Connected

AbstractThis paper is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan. We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.

scholarly journals The Fundamental Group ofS1-manifolds

2010 ◽  
Vol 62 (5) ◽  
pp. 1082-1098 ◽  
Author(s):  
Leonor Godinho ◽  
M. E. Sousa-Dias
Keyword(s):  
Fundamental Group ◽  
Moment Map ◽  
Compact Manifolds ◽  
Hamiltonian Actions

AbstractWe address the problem of computing the fundamental group of a symplecticS1-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a HamiltonianS1-action. Several examples are presented to illustrate our main results.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

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