Spherical Mean and the Fundamental Group

1991 ◽  
Vol 34 (1) ◽  
pp. 3-11 ◽  
Author(s):  
Toshiaki Adachi
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Random Walks ◽  
Isoperimetric Inequality ◽  
Lower Bounds ◽  
Compact Riemannian Manifold ◽  
Universal Covering ◽  
Spherical Means ◽  
Universal Covering Space ◽  
Spherical Mean

AbstractWe investigate some properties of spherical means on the universal covering space of a compact Riemannian manifold. If the fundamental group is amenable then the greatest lower bounds of the spectrum of spherical Laplacians are equal to zero. If the fundamental group is nontransient so are geodesic random walks. We also give an isoperimetric inequality for spherical means.

Rigidities of asymptotically Euclidean manifolds

1999 ◽  
Vol 60 (3) ◽  
pp. 521-528 ◽  
Author(s):  
Seong-Hun Paeng
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Fundamental Group ◽  
Compact Riemannian Manifold ◽  
Universal Covering ◽  
Covering Space ◽  
Universal Covering Space

Let M be an n-dimensional compact Riemannian manifold. We study the fundamental group of M when the universal covering space of M, M is close to some Euclidean space ℝs asymptotically.

On Wh3 of a Bieberbach group

1984 ◽  
Vol 95 (1) ◽  
pp. 55-60
Author(s):  
Andrew J. Nicas
Keyword(s):  
Fundamental Group ◽  
Universal Covering ◽  
Closed Manifold ◽  
Covering Space ◽  
Whitehead Group ◽  
Bieberbach Group ◽  
Universal Covering Space ◽  
Aspherical Manifold

A closed aspherical manifold is a closed manifold whose universal covering space is contractible. There is the following conjecture concerning the algebraic K-theory of such manifolds:Conjecture. Let Γ be the fundamental group of a closed aspherical manifold. Then Whi(Γ) = 0 for i ≥ 0 where Whi(Γ) is the i-th higher Whitehead group of Γ.

scholarly journals Almost Euclidean Isoperimetric Inequalities in Spaces Satisfying Local Ricci Curvature Lower Bounds

2018 ◽  
Vol 2020 (5) ◽  
pp. 1481-1510 ◽  
Author(s):  
Fabio Cavalletti ◽  
Andrea Mondino
Keyword(s):  
Riemannian Manifold ◽  
Isoperimetric Inequality ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Ricci Flow ◽  
General Framework ◽  
Isoperimetric Inequalities ◽  
Optimal Transportation ◽  
Maximal Volume ◽  
Bounded Below

Abstract Motivated by Perelman’s Pseudo-Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation.

scholarly journals On the growth of fundamental groups of nonpositive curvature manifolds

1996 ◽  
Vol 54 (3) ◽  
pp. 483-487 ◽  
Author(s):  
Yi-Hu Yang
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Sectional Curvature ◽  
Exponential Growth ◽  
Compact Riemannian Manifold ◽  
Nonpositive Curvature ◽  
Fundamental Groups ◽  
Positive Sectional Curvature ◽  
Classic Result ◽  
Negative Sectional Curvature

Milnor's classic result that the fundamental group of a compact Riemannian manifold of negative sectional curvature has exponential growth is generalised to the case of negative Ricci curvature and non-positive sectional curvature.

An infinite family of non-Haken hyperbolic 3-manifolds with vanishing Whitehead groups

1986 ◽  
Vol 99 (2) ◽  
pp. 239-246 ◽  
Author(s):  
Andrew J. Nicas
Keyword(s):  
Fundamental Group ◽  
Infinite Family ◽  
Universal Covering ◽  
Covering Space ◽  
Homotopy Equivalence ◽  
Universal Covering Space ◽  
Aspherical Manifold ◽  
Aspherical Manifolds ◽  
Whitehead Groups

A manifold M is said to be aspherical if its universal covering space is contractible. Farrell and Hsiang have conjectured [3]:Conjecture A. (Topological rigidity of aspherical manifolds.) Any homotopy equivalence f: N → M between closed aspherical manifolds is homotopic to a homeomorphism,and its analogue in algebraic K-theory:Conjecture B. The Whitehead groups Whj(π1M)(j ≥ 0) of the fundamental group of a closed aspherical manifold M vanish.

scholarly journals On 4-manifolds with universal covering space a compact geometric manifold

1993 ◽  
Vol 55 (2) ◽  
pp. 137-148
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Fundamental Group ◽  
Euler Characteristic ◽  
Homotopy Type ◽  
Universal Covering ◽  
Total Space ◽  
Covering Space ◽  
Compact Type ◽  
Universal Covering Space

AbstractThere are 11 closed 4-manifolds which admit geometries of compact type (S4, CP2 or S2 × S2) and two other closely related bundle spaces (S2 × S2 and the total space of the nontrivial RP2-bundle over S2). We show that the homotopy type of such a manifold is determined up to an ambiguity of order at most 4 by its quadratic 2-type, and this in turn is (in most cases) determined by the Euler characteristic, fundamental group and Stiefel-Whitney classes. In (at least) seven of the 13 cases, a PL 4-manifold with the same invariants as a geometric manifold or bundle space must be homeomorphic to it.

scholarly journals Energy-minimizing maps from manifolds with nonnegative Ricci curvature

2019 ◽  
pp. 1950083 ◽  
Author(s):  
James Dibble
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Universal Covering ◽  
Complete Riemannian Manifold ◽  
Conjugate Points ◽  
Nonnegative Ricci Curvature ◽  
Universal Covering Space ◽  
Asymptotic Geometry ◽  
Bounded Below ◽  
Minimizing Maps

The energy of any [Formula: see text] representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only if the original map is totally geodesic. This conclusion also holds under the weaker assumption that the domain is finitely covered by a diffeomorphic product, and its universal covering space splits isometrically as a product with a flat factor, in a commutative diagram that follows from the Cheeger–Gromoll splitting theorem.

Centralizers of reflections in crystallographic groups

1982 ◽  
Vol 92 (1) ◽  
pp. 79-91 ◽  
Author(s):  
T. Baskan ◽  
A. M. Macbeath
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Plane ◽  
Universal Covering ◽  
Interesting Case ◽  
Discrete Groups ◽  
Incompressible Surface ◽  
Crystallographic Groups ◽  
Incompressible Surfaces ◽  
Universal Covering Space ◽  
Manifold Theory

The study of hyperbolic 3-manifolds has recently been recognized as an increasingly important part of 3-manifold theory (see (9)) and for some time the presence of incompressible surfaces in a 3-manifold has been known to be important (see, for example, (4)). A particularly interesting case occurs when the incompressible surface unfolds in the universal covering space into a hyperbolic plane. The fundamental group of the surface is then contained in the stabilizer of the plane, or, what is the same thing, in the centralizer of the reflection defined by the plane. This is one motivation for studying centralizers of reflections in discrete groups of hyperbolic isometries, or, as we shall call them, hyperbolic crystallographic groups.

scholarly journals Hydrodynamic Limit of the Symmetric Exclusion Process on a Compact Riemannian Manifold

2019 ◽  
Vol 178 (1) ◽  
pp. 75-116
Author(s):  
Bart van Ginkel ◽  
Frank Redig
Keyword(s):  
Brownian Motion ◽  
Riemannian Manifold ◽  
Heat Equation ◽  
Random Walks ◽  
Hydrodynamic Limit ◽  
Compact Riemannian Manifold ◽  
Exclusion Process ◽  
Density Field ◽  
Random Grids ◽  
Empirical Density

Abstract We consider the symmetric exclusion process on suitable random grids that approximate a compact Riemannian manifold. We prove that a class of random walks on these random grids converge to Brownian motion on the manifold. We then consider the empirical density field of the symmetric exclusion process and prove that it converges to the solution of the heat equation on the manifold.

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