# A combination theorem for convex hiperbolic manifolds

Metrics details. In this paper, we prove the demiclosed principle for total asymptoticallynonexpansive nonself mappings in hyperbolic spaces. Our results extend some results inthe literature. Later, Chidume et al. Recently, Chang et al. It is well known that the demiclosed principleplays an important role in studying the asymptotic behavior for nonexpansivemappings.

Our results extend and improve the corresponding results of Chang etal. In this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [ 9 ]. Concretely, is called a hyperbolic space if is a metric space and a function satisfying. If a space satisfies only Iit coincides with the convex metric space introducedby Takahashi [ 10 ]. The concept of hyperbolic spaces in [ 9 ] is more restrictive than the hyperbolic type introduced by Goebel andKirk [ 11 ] since I - III together are equivalent to being a space of hyperbolic type in [ 11 ].

But it is slightly more general than the hyperbolic space defined inReich and Shafrir [ 12 ] see [ 9 ]. A thorough discussion of hyperbolic spaces and a detailed treatment ofexamples can be found in [ 9 ] see also [ 11 — 13 ].

A hyperbolic space is uniformly convex [ 16 ] if for, and there exists a such that. A map is called modulus of uniform convexity if for given. A subset C of a hyperbolic space X is convex if for all and. Let be a metric space and let C be a nonemptysubset of X. C is said to be a retract of Xifthere exists a continuous map such that. A map is said to be a retractionif. If P is a retraction, then for all y in the range of P. Recallthat a nonself mapping is said to be a - total asymptotically nonexpansive nonselfmapping if there exist nonnegative sequenceswith, and a strictly increasing continuous function with such that.

It iswell known that each nonexpansive mapping is an asymptotically nonexpansive mappingand each asymptotically nonexpansive mapping is a -total asymptotically nonexpansive mapping. Let be a bounded sequence in a hyperbolic space X. Forwe define.

The asymptotic radius of is given by. The asymptotic radius of with respect to is given by. The asymptotic center of is the set. The asymptotic center of with respect to is the set. Let be a complete uniformly convex hyperbolic space with monotone modulus of uniformconvexity and C a nonempty closed convex subset of X.

Then every bounded sequence in X has a unique asymptotic center with respect to C. Lemma 2 [ 17 ]. Let and be a sequence in for some. If and are sequences in X such that, and for some.

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Let, and be sequences of nonnegative numbers such that. If andthen exists. We shall prove that a total asymptotically nonexpansive nonself mapping in a completeuniformly convex hyperbolic space X with monotone modulus of uniformconvexity is demiclosed.In three-dimensional hyperbolic geometryan ideal polyhedron is a convex polyhedron all of whose vertices are ideal pointspoints "at infinity" rather than interior to three-dimensional hyperbolic space.

It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its facesmeeting along lines of the hyperbolic space. The Platonic solids and Archimedean solids have ideal versions, with the same combinatorial structure as their more familiar Euclidean versions. Several uniform hyperbolic honeycombs divide hyperbolic space into cells of these shapes, much like the familiar division of Euclidean space into cubes. However, not all polyhedra can be represented as ideal polyhedra — a polyhedron can be ideal only when it can be represented in Euclidean geometry with all its vertices on a circumscribed sphere.

Using linear programmingit is possible to test whether a given polyhedron has an ideal version, in polynomial time. Every two ideal polyhedra with the same number of vertices have the same surface area, and it is possible to calculate the volume of an ideal polyhedron using the Lobachevsky function. The surface of an ideal polyhedron forms a hyperbolic manifoldtopologically equivalent to a punctured sphere, and every such manifold forms the surface of a unique ideal polyhedron.

An ideal polyhedron can be constructed as the convex hull of a finite set of ideal points of hyperbolic space, whenever the points do not all lie on a single plane. The resulting shape is the intersection of all closed half-spaces that have the given ideal points as limit points. Alternatively, any Euclidean convex polyhedron that a circumscribed sphere [ clarification needed ] can be reinterpreted as an ideal polyhedron by interpreting the interior of the sphere as a Klein model for hyperbolic space.

Every isogonal convex polyhedron one with symmetries taking every vertex to every other vertex can be represented as an ideal polyhedron, in a way that respects its symmetries, because it has a circumscribed sphere centered at the center of symmetry of the polyhedron.

However, another highly symmetric class of polyhedra, the Catalan solidsdo not all have ideal forms. The Catalan solids are the dual polyhedra to the Archimedean solids, and have symmetries taking any face to any other face.

What is a manifold? Circle is a manifold.

Catalan solids that cannot be ideal include the rhombic dodecahedron and the triakis tetrahedron. Removing certain triples of vertices from the triakis tetrahedron separates the remaining vertices into multiple connected components. When no such three-vertex separation exists, a polyhedron is said to be 4-connected. Every 4-connected polyhedron has a representation as an ideal polyhedron; for instance this is true of the tetrakis hexahedronanother Catalan solid.

Truncating a single vertex from a cube produces a simple polyhedron one with three edges per vertex that cannot be realized as an ideal polyhedron: by Miquel's six circles theoremif seven of the eight vertices of a cube are ideal, the eighth vertex is also ideal, and so the vertices created by truncating it cannot be ideal.

There also exist polyhedra with four edges per vertex that cannot be realized as ideal polyhedra. In an ideal polyhedron, all face angles and all solid angles at vertices are zero. However, the dihedral angles on the edges of an ideal polyhedron are nonzero. The volume of an ideal tetrahedron can be expressed in terms of the Clausen function or Lobachevsky function of its dihedral angles, and the volume of an arbitrary ideal polyhedron can then be found by partitioning it into tetrahedra and summing the volumes of the tetrahedra.

The Dehn invariant of a polyhedron is normally found by combining the edge lengths and dihedral angles of the polyhedron, but in the case of an ideal polyhedron the edge lengths are infinite. This difficulty can be avoided by using a horosphere to truncate each vertex, leaving a finite length along each edge. The resulting shape is not itself a polyhedron because the truncated faces are not flat, but it has finite edge lengths, and its Dehn invariant can be calculated in the normal way, ignoring the new edges where the truncated faces meet the original faces of the polyhedron.

Because of the way the Dehn invariant is defined, and the constraints on the dihedral angles meeting at a single vertex of an ideal polyhedron, the result of this calculation does not depend on the choice of horospheres used to truncate the vertices. It can have exactly half only when the vertices can be partitioned into two equal-size independent sets, so that the graph of the polyhedron is a balanced bipartite graphas it is for an ideal cube.

Not all convex polyhedra are combinatorially equivalent to ideal polyhedra. When such an assignment exists, there is a unique ideal polyhedron whose dihedral angles are supplementary to these numbers. As a consequence of this characterization, realizability as an ideal polyhedron can be expressed as a linear program with exponentially many constraints one for each non-facial cycleand tested in polynomial time using the ellipsoid algorithm.Springer Shop Labirint Ozon.

Hyperbolic Manifolds and Discrete Groups. Michael KapovichMikhail Kapocivh. This important work is at the crossroads of several branches of mathematics, including hyperbolic geometry, discrete groups, 3-dimensional topology, geometric group theory, and complex analysis.

The main focus throughout is on the 'Big Monster,' that is, on Thurston's Hyperbolization Theorem, which has completely changed the landscape of 3-dimensional topology and Kleinian group theory.

Presented in the work is the first complete proof of Thurston's Hyperbolization Theorem in the 'generic case'. The author begins with an outline of each chapter in a detailed introduction, which also contains a number of open problems and conjectures in Section 1. Further topics are laid out in systematic fashion, including an extended treatment of the theory of Kleinian groups and group actions on trees.

Also presented are several deep theorems and their proofs about 3-manifolds and discrete groups: the Haken hierarchy theorem, Waldhausen's homeomorphism theorem, Bonahon's theorem, Ahlfors finiteness theorem, and Sullivan's rigidity theorem. Hyperbolic Manifolds and Discrete Groups is replete with beautiful illustrations and examples of key concepts.

It should serve as a comprehensive reference for mathematicians and students or as a text for a graduate seminar. Series: Progress in Mathematics, Vol. Introduction 1. Three-dimensional Topology 2. Thurston Norm 3. Geometry of the Hyperbolic Space 4. Teichmueller Theory of Riemannian Surfaces 5.

Kleinian Groups 6. Introduction to the Orbifold Theory 7. Complex Projective Structures 8. Sociology of Kleinian Groups 9.We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold.

In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold. We study the following general question. All the manifolds in this paper are assumed implicitly to be smooth, connected, and oriented, unless otherwise stated. Let M be a compact smooth n -manifold with non-empty boundary.

Is there a closed hyperbolic n -manifold W containing M as a convex submanifold? Here convex means that every arc in M is homotopic relative to its endpoints in M to a geodesic. This is in fact a local property of the boundary of Msince M is convex if and only if it is locally convex [ 2Section 1.

This is a manifestation of geometrisation, see Remark 9. One motivation is that being convex gives M some privileges.

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So in particular the interior of M has a geometrically finite hyperbolic structure that covers the closed hyperbolic W. See Sect.

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The main contribution of this paper is to furnish a family of examples in dimension 4. Loops and multiple edges connecting two nodes are allowed.

See [ 3Section 4. The regular neighbourhood of a generically immersed closed possibly disconnected surface in a four-manifold is a disjoint union of plumbings.

Plumbings are ubiquitous in dimension four and it is natural to ask whether they can be embedded as convex subsets in some closed hyperbolic four-manifold. We prove here the following. The manifold M is contained as a convex subset in a closed hyperbolic four-manifold W with. This result is already new to the best of our knowledge when the plumbing graph is a point.

Theorem 2 implies the following. For every symmetric integer matrix Q there is a boundary connected sum of plumbings M with intersection form Q that embeds as a convex submanifold into a closed hyperbolic 4-manifold W. We get a plumbing M with intersection form Q that embeds convexly in a closed hyperbolic 4-manifold.

For every symmetric integer matrix Q there is a boundary connected sum of plumbings M with intersection form Qwhose interior admits a geometrically finite complete hyperbolic structure that covers a closed hyperbolic manifold W. All these manifolds M and W are constructed by assembling right-angled cells. We find here many examples of non-spin closed hyperbolic 4-manifolds W. We constructed the first such manifolds recently in [ 14 ], and the techniques employed here are an extension of these.

Such manifolds M exist in all dimensions, also with connected boundary [ 10 ]. Some explicit examples were constructed in [ 8 ] using the right-angled cell.We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups, there are finite index subgroups which generate a subgroup that is an amalgamated free product.

Constructions of infinite volume hyperbolic n-manifolds are described by gluing lower dimensional manifolds. It is shown that every slope on a cusp of a hyperbolic 3-manifold is a multiple immersed boundary slope. If a 3-manifold contains a maximal surface group not carried by an embedded surface then it contains the fundamental group of a book of I-bundles with more than two pages.

Documents: Advanced Search Include Citations. DMCA A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds Cached Download Links [arxiv.

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Citations: 8 - 1 self. Abstract We prove the convex combination theorem for hyperbolic n-manifolds. Powered by:.LIVE CHAT CONTACT US LIVE CHAT CONTACT US FEEDBACK. We know that consumers look to friends, family, and even strangers for feedback and recommendations on services and products. They trust the word of mouth marketing. We use the internet to research before we buy, so what people say online matters and influence whether or not someone purchases from you.

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### Ideal polyhedron

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### Convex plumbings in closed hyperbolic 4-manifolds

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