Bruce Kleiner
Courant Institute of Mathematical Sciences
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Geometry & Topology | 2008
Bruce Kleiner; John Lott
e Conjecture, and more generally Thurston’s Geometrization Conjecture, using the Ricci flow approach of Hamilton. Perelman’s proofs are concise and, at times, sketchy. The purpose of these notes is to provide the details that are missing in [46] and [47], which contain Perelman’s arguments for the Geometrization Conjecture. Among other things, we cover the construction of the Ricci flow with surgery of [47]. We also discuss the long-time behavior of the Ricci flow with surgery, which is needed for the full Geometrization Conjecture. The papers of Colding and Minicozzi [23; 24] and Perelman [48], which are not covered in these notes, each provide a shortcut in the case of the Poincar´ e Conjecture. Namely, these papers show that if the initial manifold is simply-connected then the Ricci flow with surgery becomes extinct in a finite time, thereby removing the issue of the long-time behavior. Combining this claim with the proof of existence of Ricci flow with surgery gives the shortened proof in the simply-connected case. These notes are intended for readers with a solid background in geometric analysis. Good sources for background material on Ricci flow are Chow and Knopf [21], Chow, Lu and Ni [22], Hamilton [30] and Topping [60]. The notes are self-contained but are designed to be read along with [46; 47]. For the most part we follow the format of [46; 47] and use the section numbers of [46; 47] to label our sections. We have done this in
Mathematische Zeitschrift | 1999
Bruce Kleiner
Abstract. We show that a number of different notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT(0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are determined by the maximal dimension of a flat in X.
Inventiones Mathematicae | 2002
Mario Bonk; Bruce Kleiner
We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standard 2-sphere, answering a question of Heinonen and Semmes.
Topology | 2000
Christopher B. Croke; Bruce Kleiner
Abstract We construct a pair of finite piecewise Euclidean 2-complexes with nonpositive curvature which are homeomorphic but whose universal covers have nonhomeomorphic ideal boundaries, settling a question of Gromov.
Journal of the American Mathematical Society | 2010
Bruce Kleiner
We give a new proof of Gromov’s theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. The proof does not rely on the MontgomeryZippin-Yamabe structure theory of locally compact groups.
Annales Scientifiques De L Ecole Normale Superieure | 2000
Michael Kapovich; Bruce Kleiner
If a torsion-free hyperbolic group G has 1-dimensional boundary ∂∞G, then ∂∞G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When ∂∞G is a Sierpinski carpet we show that G is a quasi-convex subgroup of a 3-dimensional hyperbolic Poincare duality group. We also construct a “topologically rigid” hyperbolic group G: any homeomorphism of ∂∞G is induced by an element of G.
Geometry & Topology | 2005
G. Christopher Hruska; Bruce Kleiner
We explore the geometry of nonpositively curved spaces with isolated flats, and its consequences for groups that act properly discontinuously, cocompactly, and isometrically on such spaces. We prove that the geometric boundary of the space is an invariant of the group up to equivariant homeomorphism. We also prove that any such group is relatively hyperbolic, biautomatic, and satisfies the Tits Alternative. The main step in establishing these results is a characterization of spaces with isolated flats as relatively hyperbolic with respect to flats. Finally we show that a CAT(0) space has isolated flats if and only if its Tits boundary is a disjoint union of isolated points and standard Euclidean spheres. In an appendix written jointly with Hindawi, we extend many of the results of this article to a more general setting in which the isolated subspaces are not required to be flats. AMS Classification numbers Primary: 20F67 Secondary: 20F69
Geometry & Topology | 2005
Mario Bonk; Bruce Kleiner
Suppose G is a Gromov hyperbolic group, and @1G is quasisymmetrically homeomorphic to an Ahlfors Q–regular metric 2–sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on H 3 .
Geometric and Functional Analysis | 2009
Jeff Cheeger; Bruce Kleiner
We prove the differentiability of Lipschitz maps X → V, where X denotes a PI space, i.e. a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon–Nikodym Property (RNP). As a consequence, we obtain a bi-Lipschitz nonembedding theorem for RNP targets. The differentiation theorem depends on a new specification of the differentiable structure for PI spaces involving directional derivatives in the direction of velocity vectors to rectifiable curves. We give two different proofs of this, the second of which relies on a new characterization of the minimal upper gradient. There are strong implications for the infinitesimal structure of PI spaces which will be discussed elsewhere.
Geometry & Topology | 2008
Mladen Bestvina; Bruce Kleiner; Michah Sageev
We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset Q which is locally flat outside a compact set, and asymptotically conical.