相关论文: Warped cones, (non-)rigidity, and piecewise proper…

In this article, we prove that if two warped cones corresponding to two finitely generated groups with free, isometric, measure-preserving, actions on two compact metric spaces with probability measures are level-wise quasi-isometric (with…

We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian…

Warped cones are metric spaces introduced by John Roe from discrete group actions on compact metric spaces to produce interesting examples in coarse geometry. We show that a certain class of warped cones $\mathcal{O}_\Gamma (M)$ admit a…

The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real…

It is known that the expanders arising as increasing sequences of level sets of warped cones, as introduced by the second-named author, do not coarsely embed into a Banach space as soon as the corresponding warped cone does not coarsely…

We introduce cone bilipschitz equivalences between metric spaces. These are maps, more general than quasi-isometries, that induce a bilipschitz homeomorphism between asymptotic cones. Non-trivial examples appear in the context of Lie…

In the present paper we prove lemmata on strong contractibility in asymptotic cones and metric ultraproducts which we apply to both the case of finitely generated word norms and the case of conjugation invariant norms. We recover…

In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by St{\o}rmer. Our method is based on a definition of an inner product in the space of linear maps between two…

We construct metric spaces that do not have property A yet are coarsely embeddable into the Hilbert space. Our examples are so called warped cones, which were introduced by J. Roe to serve as examples of spaces non-embeddable into a Hilbert…

We compute the covering dimension the asymptotic cone of a connected Lie group. For simply connected solvable Lie groups, this is the codimension of the exponential radical. As an application of the proof, we give a characterization of…

We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold. This extends the classical notion of asymptotic directions usually defined on smooth submanifolds. We…

Given a bi-invariant metric on a group, we construct a version of an asymptotic cone without using ultrafilters. The new construction, called the directional asymptotic cone, is a contractible topological group equipped with a complete…

In this paper, we first prove that any power quasi-symmetry of two metric spaces induces a rough quasi-isometry between their infinite hyperbolic cones. Second, we prove that for a complete metric space $Z$, there exists a point $\omega$ in…

Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are…

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, rank 1 CAT(0)…

We show that warped cones over actions with spectral gaps do not embed coarsely into large classes of Banach spaces. In particular, there exist warped cones over actions of the free group that do not embed coarsely into $L_p$-spaces and…

This paper completes a fundamental construction in Alexandrov geometry. Previously we gave a new construction of metric spaces with curvature bounds either above or below, namely warped products with intrinsic metric space base and fiber,…

We prove that complete warped product Einstein metrics with isometric bases, simply connected space form fibers, and the same Ricci curvature and dimension are isometric. In the compact case we also prove that the warping functions must be…

We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the…

We define a way of approximating actions on measure spaces using finite graphs; we then show that in quite general settings these graphs form a family of expanders if and only if the action is expanding in measure. This provides a somewhat…