Manifolds of Nonpositive Curvature

Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor

doi:10.1007/978-1-4684-9159-3

Cited by 559 publications

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“…The first part of (ii) follows easily from Toponogov's theorem in the non-positive case, see [4]. That the angles approach zero as r approaches infinity follows from the hyperbolic cosine rule, see [5].…”

Section: Iii) Area (T(\si \S 2 \S 3 )) Is An Increasing Function Omentioning

confidence: 97%

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Singular pleated surfaces and CP¹–structures

Tan¹

1995

Glasgow Math. J.

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1. Introduction. Let F g be a closed orientable surface of genus g > 1 and let ST g be the Teichmuller space of F g , i.e., the space of marked hyperbolic structures on F g . We shall also denote by ST gA the space of marked hyperbolic structures on F s with one distinguished point; by this, we mean a distinguished point on the universal cover F g of F g . This space is isomorphic to the space of marked complete hyperbolic structures on a genus g surface with 1 cusp which is the usual interpretation of ST giX . Choose a decomposition of F g into pairs of pants by a collection of non-intersecting, totally geodesic simple closed curves. The Fenchel-Nielsen coordinates for ST g relative to this decomposition are given by the lengths of the curves as well as twist parameters defined on each curve. Varying the length and twist parameters gives deformations of the marked hyperbolic structures.To obtain deformations of the CP

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Section: Iii) Area (T(\si \S 2 \S 3 )) Is An Increasing Function Omentioning

confidence: 97%

“…negative directions relative to a fixed orientation of the surface. 4. As e increases, the curvature of the singular point increases (equivalently, the cone angle decreases).…”

Section: Coordinates For Cp'-structuresmentioning

confidence: 98%

Singular pleated surfaces and CP¹–structures

Tan¹

1995

Glasgow Math. J.

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“…In [1], it was shown that for a Hadamard manifold every horofunction is the Busemann-function, and hence the ideal boundary can be defined by the equivalence classes of rays emanating from any fixed point of M. The following theorem is a generalization of this fact for surfaces admitting total curvature. One of our main objects is to prove this theorem, which will be stated again later.…”

Section: Convexity Of the Ideal Boundarymentioning

confidence: 99%

“…We will follow the lead of [1], [6]. Let M be a noncompact complete Riemannian manifold with distance function d( , ).…”

Section: Introductionmentioning

confidence: 99%

Convexity of the ideal boundary for complete open surfaces

Yim¹

1995

Trans. Amer. Math. Soc.

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“…See also the recent review article by Kleiner in the Bulletin of the AMS [59] and the related books by Ballmann [4], BallmannGromov-Schroeder [5] and the original long article by Gromov [48]. In this article I focus not on the consequences of the theory, but rather on the question of how to construct examples to which it applies.…”

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confidence: 99%

Constructing non-positively curved spaces and groups

McCammond¹

2009

Geometric and Cohomological Methods in Group Theory

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Abstract. The theory of non-positively curved spaces and groups is tremendously powerful and has enormous consequences for the groups and spaces involved. Nevertheless, our ability to construct examples to which the theory can be applied has been severely limited by an inability to test -in real timewhether a random finite piecewise Euclidean complex is non-positively curved. In this article I focus on the question of how to construct examples of nonpositively curved spaces and groups, highlighting in particular the boundary between what is currently do-able and what is not yet feasible. Since this is intended primarily as a survey, the key ideas are merely sketched with references pointing the interested reader to the original articles.Over the past decade or so, the consequences of non-positive curvature for geometric group theorists have been throughly investigated, most prominently in the book by Bridson and Haefliger [26]. See also the recent review article by Kleiner in the Bulletin of the AMS [59] and the related books by Ballmann [4], BallmannGromov-Schroeder [5] and the original long article by Gromov [48]. In this article I focus not on the consequences of the theory, but rather on the question of how to construct examples to which it applies. The structure of the article roughly follows the structure of the lectures I gave during the Durham symposium with the four parts corresponding to the four talks. Part 1 introduces the key problems and presents some basic decidability results, Part 2 focuses on practical algorithms and low-dimensional complexes, Part 3 presents case studies involving special classes of groups such as Artin groups, small cancellation groups and ample-twisted face pairing 3-manifolds. In Part 4 I explore the weaker notion of conformal non-positive curvature and introduce the notion of an angled n-complex. As will become clear, the topics covered have a definite bias towards research in which I have played some role. These are naturally the results with which I am most familiar and I hope the reader will pardon this lack of a more impartial perspective. Part 1. Negative curvatureAlthough the definitions of δ-hyperbolic and CAT(0) spaces / groups are wellknown, it is perhaps under-appreciated that there are at least seven potentially distinct classes of groups which all have some claim to the description "negativelycurved." Sections 1 and 2 briefly review the definitions needed in order to describe these classes, the known relationships between them, and the current status of the (rather optimisitic) conjecture that all seven of these classes are identical. Since the emphasis in this article is on the construction of examples, finite piecewise

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Manifolds of Nonpositive Curvature

Cited by 559 publications

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Singular pleated surfaces and CP¹–structures

Singular pleated surfaces and CP¹–structures

Convexity of the ideal boundary for complete open surfaces

Constructing non-positively curved spaces and groups

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Manifolds of Nonpositive Curvature

Cited by 559 publications

References 0 publications

Singular pleated surfaces and CP1–structures

Singular pleated surfaces and CP1–structures

Convexity of the ideal boundary for complete open surfaces

Constructing non-positively curved spaces and groups

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Singular pleated surfaces and CP¹–structures

Singular pleated surfaces and CP¹–structures