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SURVEY ON ASPHERICAL MANIFOLDS
, 2009
"... 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 nonpositive curvature conditions. The property aspherical, which is a purely homotopy theoreti ..."
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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 nonpositive 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 Ltheory 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.
C ∗SIMPLE GROUPS: AMALGAMATED FREE PRODUCTS, HNN EXTENSIONS, AND FUNDAMENTAL GROUPS OF 3MANIFOLDS
, 909
"... Abstract. We establish sufficient conditions for the C ∗simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNNextensions, and their normal subgroups; for example normal subgroups of BaumslagSolitar groups. The second class is ..."
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Abstract. We establish sufficient conditions for the C ∗simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNNextensions, and their normal subgroups; for example normal subgroups of BaumslagSolitar groups. The second class is that of fundamental groups of compact 3manifolds, related to the first class by their KneserMilnor and JSJdecompositions. Much of our analysis deals with conditions on an action of a group Γ on a tree T which imply the following three properties: abundance of hyperbolic elements, better called strong hyperbolicity, minimality, both on the tree T and on its boundary ∂T, and faithfulness in a strong sense. For this, we define in particular the notion of a slender automorphism of T, namely of an automorphism such that its set of fixed points on ∂T is nowhere dense with respect to the shadow topology. 1.
Systoles and Dehn surgery for hyperbolic 3manifolds
"... Abstract Given a closed hyperbolic 3manifold M of volume V , and a link L ⊂ M such that the complement M \ L is hyperbolic, we establish a bound for the systole length of M \ L in terms of V . This extends a result of Adams and Reid, who showed that in the case that M is not hyperbolic, there is a ..."
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Abstract Given a closed hyperbolic 3manifold M of volume V , and a link L ⊂ M such that the complement M \ L is hyperbolic, we establish a bound for the systole length of M \ L in terms of V . This extends a result of Adams and Reid, who showed that in the case that M is not hyperbolic, there is a universal bound of 7.35534... . As part of the proof, we establish a bound for the systole length of a noncompact finite volume hyperbolic manifold which grows asymptotically like 4 3 log V .
Systoles and Dehn surgery for hyperbolic 3–manifolds.
, 2013
"... Given a Riemannian n–manifold M the systole of M is defined to be the shortest closed geodesic on M, and the systole length sys(M) provides an interesting invariant of M. Gromov [13] proved that in each dimension n, there is a universal constant Cn such that ..."
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Given a Riemannian n–manifold M the systole of M is defined to be the shortest closed geodesic on M, and the systole length sys(M) provides an interesting invariant of M. Gromov [13] proved that in each dimension n, there is a universal constant Cn such that
Minimal volume and simplicial norm of visibility nmanifolds and compact 3manifolds
, 2009
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Long time behaviour of Ricci flow on open 3manifolds
, 2012
"... We study the long time behaviour of Ricci flow with bubblingoff on a possibly noncompact 3manifold of finite volume whose universal cover has bounded geometry. As an application, we give a Ricci flow proof of Thurston’s hyperbolisation theorem for 3manifolds with toral boundary that generalizes ..."
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We study the long time behaviour of Ricci flow with bubblingoff on a possibly noncompact 3manifold of finite volume whose universal cover has bounded geometry. As an application, we give a Ricci flow proof of Thurston’s hyperbolisation theorem for 3manifolds with toral boundary that generalizes Perelman’s proof of the hyperbolisation conjecture in the closed case. 1