Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

Belegradek, Igor

doi:10.2140/agt.2006.6.1341

Cited by 10 publications

(11 citation statements)

References 34 publications

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“…Belegradek in [Bel06] asked if these homology spheres have positive simplicial volume. Our main theorem answers this question affirmatively, but the main part of our argument is to do with the upper bound on simplicial volume.…”

Section: Introductionmentioning

confidence: 99%

“…They seem to be the first exmples of 4-dimensional homology spheres with Riemannian metrics of non-positive sectional curvature (see the paragraph after Remark 5 in [NW00]). Belegradek in [Bel06] asked if these homology spheres have positive simplicial volume. Our main theorem answers this question affirmatively, but the main part of our argument is to do with the upper bound on simplicial volume.…”

Section: Introductionmentioning

confidence: 99%

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Simplicial volume and fillings of hyperbolic manifolds

Fujiwara

Manning

2011

Algebr. Geom. Topol.

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Let M be a hyperbolic n-manifold whose cusps have torus cross-sections. In [8], the authors constructed a variety of nonpositively and negatively curved spaces as "2 -fillings" of M by replacing the cusps of M with compact "partial cones" of their boundaries. These 2 -fillings are closed pseudomanifolds, and so have a fundamental class. We show that the simplicial volume of any such 2 -filling is positive, and bounded above by, where v n is the volume of a regular ideal hyperbolic n-simplex. This result generalizes the fact that hyperbolic Dehn filling of a 3-manifold does not increase hyperbolic volume.In particular, we obtain information about the simplicial volumes of some 4-dimensional homology spheres described by Ratcliffe and Tschantz, answering a question of Belegradek and establishing the existence of 4-dimensional homology spheres with positive simplicial volume.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Simplicial volume and fillings of hyperbolic manifolds

Fujiwara

Manning

2011

Algebr. Geom. Topol.

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“…Furthermore, without aiming at being exhaustive, here we just mention that Gromov Additivity Theorem has been also exploited in studying the possible degrees of maps between manifolds [42,49,17,8,18,19], in establishing results about the behavior of manifolds under collapse [7,5], and in various other areas of low-dimensional topology [1,38,6,4,25,47,34,30,3].…”

Section: Introductionmentioning

confidence: 99%

Isometric embeddings in bounded cohomology

et al. 2014

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This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X,Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X,Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along \pi_1-injective boundary components with amenable fundamental group

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“…This applies for instance to the Kaplansky Conjecture, Unit Conjecture, Zero-divisor-conjecture, Baum-Connes Conjecture, Farrell-Jones Conjecture for algebraic K-theory for regular R, Farrell-Jones Conjecture for algebraic L-theory, the vanishing of K 0 (ZG) and of Wh(G) = 0, For information about these conjectures and their links we refer for instance to [6], [68] and [70]. Further similar consequences of the reflection group trick can be found in Belegradek [8].…”

Section: Hyperbolizationmentioning

confidence: 80%

“…Notice that a finitely presented group with unsolvable word problem is not a CAT(0)-group, not hyperbolic, not automatic, not asynchronously automatic, not residually finite and not linear over any commutative ring (see [8,Remark 5.2]).…”

Section: Hyperbolizationmentioning

confidence: 99%

Survey on aspherical manifolds

Lück¹

European Congress of Mathematics Amsterdam, 14–18 July, 2008

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Abstract. This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical, which is a purely homotopy theoretical condition, implies many striking results about the geometry and analysis of the manifold or its universal covering, and the ring theoretic properties and the K-and L-theory of the group ring associated to its fundamental group. The Borel Conjecture predicts that closed aspherical manifolds are topologically rigid. The article contains new results about product decompositions of closed aspherical manifolds and an announcement of a result joint with Arthur Bartels and Shmuel Weinberger about hyperbolic groups with spheres of dimension ≥ 6 as boundary. At the end we describe (winking) our universe of closed manifolds. IntroductionA space X is called aspherical if it is path connected and all its higher homotopy groups vanish, i.e., π n (X) is trivial for n ≥ 2. This survey article is devoted to aspherical closed manifolds. These are very interesting objects for many reasons. Often interesting geometric constructions or examples lead to aspherical closed manifolds. The study of the question which groups occur as fundamental groups of closed aspherical manifolds is intriguing. The condition aspherical is of purely homotopy theoretical nature. Nevertheless there are some interesting questions and conjectures about curvature properties of a closed aspherical Riemann manifold and about the spectrum of the Laplace operator on its universal covering. The Borel Conjecture predicts that aspherical closed topological manifolds are topologically rigid and that aspherical compact Poincaré complexes are homotopy equivalent to closed manifolds. We discuss the status of some of these questions and conjectures. Examples of exotic aspherical closed manifolds come from hyperbolization techniques and we list certain examples. At the end we describe (winking) our universe of closed manifolds.The results about product decompositions of closed aspherical manifolds in Section 6 are new and Section 8 contains an announcement of a result joint with Arthur Bartels and Shmuel Weinberger about hyperbolic groups with spheres of dimension ≥ 6 as boundary.

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Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

Cited by 10 publications

References 34 publications

Simplicial volume and fillings of hyperbolic manifolds

Simplicial volume and fillings of hyperbolic manifolds

Isometric embeddings in bounded cohomology

Survey on aspherical manifolds

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