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Bruce Kleiner

Publications and source records attributed to Bruce Kleiner.

At least 55 records · Page 3Linked to original sources
arXiv2011

Rigidity of Schottky sets

We call a complement of a union of at least three disjoint (round) open balls in the unit sphere S^n a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a Mobius transformation on S^n. In the other direction we show that every Schottky set in S^2 of positive measure admits non-trivial quasisymmetric maps to other Schottky sets. These results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries.

math.MG↗
arXiv2010

Geometry and rigidity of mapping class groups

We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namely that groups quasi-isometric to MCG(S) are virtually equal to it. (The latter theorem was proved by Hamenstadt using different methods). As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG(S); a characterization of the image of the curve-complex projection map from MCG(S) to the product of the curve complexes of essential subsurfaces of S; and a construction of Sigma-hulls in MCG(S), an analogue of convex hulls.

math.GT↗
arXiv2009

A $(\log n)^{Ω(1)}$ integrality gap for the Sparsest Cut SDP

We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap $(\log n)^{Ω(1)}$. This is achieved by exhibiting $n$-point metric spaces of negative type whose $L_1$ distortion is $(\log n)^{Ω(1)}$. Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to $L_1$ when restricted to cosets of the center.

cs.DS↗
arXiv2009

Metric differentiation, monotonicity and maps to L^1

We give a new approach to the infinitesimal structure of Lipschitz maps into L^1. As a first application, we give an alternative proof of the main theorem from an earlier paper, that the Heisenberg group does not admit a bi-Lipschitz embedding in L^1. The proof uses the metric differentiation theorem of Pauls and the cut metric decomposition to reduce the nonembedding argument to a classification of monotone subsets of the Heisenberg group.

math.MG↗
arXiv2008

Hadamard spaces with isolated flats

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.

math.GR↗
arXiv2008

Erratum on "Hadamard spaces with isolated flats"

The purpose of this erratum is to correct the proof of Theorem A.0.1 in the appendix to our article ``Hadamard spaces with isolated flats'' math.GR/0411232, which was jointly authored by Mohamad Hindawi, Hruska and Kleiner. In that appendix, many of the results of math.GR/0411232 about CAT(0) spaces with isolated flats are extended to a more general setting in which the isolated subspaces are not necessarily flats. However, one step of that extension does not follow from the argument we used the isolated flats setting. We provide a new proof that fills this gap. In addition, we give a more detailed account of several other parts of Theorem A.0.1, which were sketched in math.GR/0411232.

math.GR↗
arXiv2008

Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon Nikodym property

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direction of tangent vectors to suitable rectifiable curves.

math.MG↗
arXiv2008

Induced quasi-actions: a remark

In this note we observe that the notion of an induced representation has an analog for quasi-actions. We then use induced quasi-actions to refine some earlier rigidity results for product spaces.

math.GR↗
arXiv2007

A new proof of Gromov's theorem on groups of polynomial growth

We give a new proof of Gromov's theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.

math.GR↗
arXiv2007

The asymptotic geometry of right-angled Artin groups, I

We study atomic right-angled Artin groups -- those whose defining graph has no cycles of length less than five, and no separating vertices, separating edges, or separating vertex stars. We show that these groups are not quasi-isometrically rigid, but that an intermediate form of rigidity does hold. We deduce from this that two atomic groups are quasi-isometric iff they are isomorphic.

math.GR↗
arXiv2006

Differentiating maps into L^1 and the geometry of BV functions

This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps, X-->V, and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V=L^1 where differentiability fails. We establish another kind of differentiability for certain X, including R^n and H, the Heisenberg group with its Carnot-Cartheodory metric. It follows that H does not bi-Lipschitz embed into L^1, as conjectured by J. Lee and A. Naor. When combined with their work, this provides a natural counter example to the Goemans-Linial conjecture in theoretical computer science; the first such counterexample was found by Khot-Vishnoi. A key ingredient in the proof of our main theorem is a new connection between Lipschitz maps to L^1 and functions of bounded variation, which permits us to exploit recent work on the structure of BV functions on the Heisenberg group.

math.MG↗
arXiv2003

Singularity structure in mean curvature flow of mean convex sets

In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres.

math.DG↗
arXiv2001

Rigidity for Quasi-Mobius group actions

Suppose G is a hyperbolic group whose boundary has topological dimension k. If the boundary is quasisymmetrically homeomorphic to an Ahlfors k-regular metric space, then, modulo a finite normal subgroup, G is isomorphic to a uniform lattice in the isometry group of hyperbolic (k+1)-space.

math.MG↗