EXPLICIT CONSTRUCTIONS OF UNIVERSAL ℝ-TREES AND ASYMPTOTIC GEOMETRY OF HYPERBOLIC SPACES

2001 ◽  
Vol 33 (6) ◽  
pp. 727-734 ◽  
Author(s):  
ANNA DYUBINA ◽  
IOSIF POLTEROVICH
Keyword(s):  
Free Group ◽  
Negative Curvature ◽  
Axiom Of Choice ◽  
Hyperbolic Spaces ◽  
Asymptotic Cone ◽  
Connected Manifold ◽  
Explicit Constructions ◽  
The Axiom Of Choice ◽  
Asymptotic Geometry ◽  
Simply Connected

This paper presents explicit constructions of universal ℝ-trees as certain spaces of functions, and also proves that a 2ℵ0-universal ℝ-tree can be isometrically embedded at infinity into a complete simply connected manifold of negative curvature, or into a non-abelian free group. In contrast to asymptotic cone constructions, asymptotic spaces are built without using the axiom of choice.

scholarly journals A Statistical Cohomogeneity One Metric on the Upper Plane with Constant Negative Curvature

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Limei Cao ◽  
Didong Li ◽  
Erchuan Zhang ◽  
Zhenning Zhang ◽  
Huafei Sun
Keyword(s):  
Vector Field ◽  
Hyperbolic Space ◽  
Negative Curvature ◽  
Statistical Manifold ◽  
Constant Negative Curvature ◽  
Cohomogeneity One ◽  
Geometrical Structures ◽  
Connected Manifold ◽  
Jacobi Vector ◽  
Simply Connected

we analyze the geometrical structures of statistical manifoldSconsisting of all the wrapped Cauchy distributions. We prove thatSis a simply connected manifold with constant negative curvatureK=-2. However, it is not isometric to the hyperbolic space becauseSis noncomplete. In fact,Sis approved to be a cohomogeneity one manifold. Finally, we use several tricks to get the geodesics and explore the divergence performance of them by investigating the Jacobi vector field.

Subgroups of a free group and the axiom of choice

1985 ◽  
Vol 50 (2) ◽  
pp. 458-467 ◽  
Author(s):  
Paul E. Howard
Keyword(s):  
Free Group ◽  
Cardinal Number ◽  
Cyclic Permutation ◽  
Axiom Of Choice ◽  
Embedding Theorem ◽  
Open Problems ◽  
Power Set ◽  
Partial List ◽  
Image Position ◽  
The Axiom Of Choice

Nielsen [7] has proved that every subgroup of a free group of finite rank is free. The theorem was later strengthened by Schreier [8] by eliminating the finiteness restriction on the rank. Several proofs of this theorem (known as the Nielsen-Schreier theorem, henceforth denoted by NS) have appeared since Schreier's 1927 paper (see [1] and [2]). All proofs of NS use the axiom of choice (AC) and it is natural to ask whether NS is equivalent to AC. Läuchli has given a partial answer to this question by proving [6] that the negation of NS is consistent with ZFA (Zermelo-Fraenkel set theory weakened to permit the existence of atoms). By the Jech-Sochor embedding theorem (see [3] and [4]) ZFA can be replaced by ZF. Some form of AC, therefore, is needed to prove NS. The main purpose of this paper is to give a further answer to this question.In §2 we prove that NS implies ACffin (the axiom of choice for sets of finite sets). In §3 we show that a strengthened version of NS implies AC and in §4 we give a partial list of open problems.Let y be a set; ∣y∣ denotes the cardinal number of y and (y) is the power set of y. If p is a permutation of y and t ∈ y, the p-orbit of t is the set {pn(t): n is an integer}. Ifwe call p a cyclic permutation of y. If f is a function with domain y and x ⊆ y, f″x denotes the set {f(t):t ∈ x}. If A is a subset of a group (G, °) (sometimes (G, °) will be denoted by G) then A−1 = {x−1:x ∈ A} and [A] denotes the subgroup of G generated by A.

The Axiom of Choice Machine

2018 ◽  
Author(s):  
Alexander R. Pruss
Keyword(s):  
Set Theory ◽  
Laws Of Nature ◽  
Axiom Of Choice ◽  
Dutch Book ◽  
The Axiom Of Choice

This is a mainly technical chapter concerning the causal embodiment of the Axiom of Choice from set theory. The Axiom of Choice powered a construction of an infinite fair lottery in Chapter 4 and a die-rolling strategy in Chapter 5. For those applications to work, there has to be a causally implementable (though perhaps not compatible with our laws of nature) way to implement the Axiom of Choice—and, for our purposes, it is ideal if that involves infinite causal histories, so the causal finitist can reject it. Such a construction is offered. Moreover, other paradoxes involving the Axiom of Choice are given, including two Dutch Book paradoxes connected with the Banach–Tarski paradox. Again, all this is argued to provide evidence for causal finitism.

Torus actions, maximality, and non-negative curvature

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Christine Escher ◽  
Catherine Searle
Keyword(s):  
Negative Curvature ◽  
Torus Action ◽  
Maximal Symmetry ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Product Of Spheres ◽  
Rank Conjecture ◽  
Simply Connected ◽  
Symmetry Rank ◽  
Special Case

Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.

scholarly journals Yang–Mills theory and jumping curves

2015 ◽  
Vol 26 (05) ◽  
pp. 1550029
Author(s):  
Yasha Savelyev
Keyword(s):  
Vector Bundle ◽  
Holomorphic Vector Bundle ◽  
Chain Complex ◽  
Complex Manifolds ◽  
Smooth Manifolds ◽  
Hermitian Vector Bundle ◽  
Connected Manifold ◽  
Classical Sense ◽  
Yang Mills ◽  
Simply Connected

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang–Mills theory over S2 to show that any non-trivial, smooth Hermitian vector bundle E over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-trivial holomorphic vector bundle must have jumping curves in the classical sense (when c1(E) is zero). We also use this to give a new proof of a theorem of Gromov on the norm of curvature of unitary connections, and make the theorem slightly sharper. Lastly we define a sequence of new non-trivial integer invariants of smooth manifolds, connected to this theory of smooth jumping curves, and make some computations of these invariants. Our methods include an application of the recently developed Morse–Bott chain complex for the Yang–Mills functional over S2.

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