Straightening warped cones

We provide the converses to two results of Roe [Warped cones and property A, Geom. Topol. 9 (2005) 163–178, https://doi.org/10.2140/9t.2005.9.163 ]: first, the warped cone associated to a free action of an a-T-menable group admits a fibered coarse embedding into a Hilbert space, and second, a free action yielding a warped cone with property A must be amenable. We construct examples showing that in both cases the freeness assumption is necessary. The first equivalence is obtained also for other classes of Banach spaces, in particular for [Formula: see text]-spaces.

Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings

The objective of this series is to study metric geometric properties of disjoint unions of Cayley graphs of amenable groups by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint union admits a fibred coarse embedding into a Hilbert space (as a disjoint union) if and only if the Cayley boundary of the sequence in the space of marked groups is uniformly a-T-menable. We furthermore extend this result to ones with other target spaces. By combining our main results with constructions of Osajda and Arzhantseva–Osajda, we construct two systems of markings of a certain sequence of finite groups with two opposite extreme behaviors of the resulting two disjoint unions: With respect to one marking, the space has property A. On the other hand, with respect to the other, the space does not admit fibred coarse embeddings into Banach spaces with non-trivial type (for instance, uniformly convex Banach spaces) or Hadamard manifolds; the Cayley limit group is, furthermore, non-exact.

From the geometry of box spaces to the geometry and measured couplings of groups

In this paper, we prove that if two “box spaces” of two residually finite groups are coarsely equivalent, then the two groups are “uniform measured equivalent” (UME). More generally, we prove that if there is a coarse embedding of one box space into another box space, then there exists a “uniform measured equivalent embedding” (UME-embedding) of the first group into the second one. This is a reinforcement of the easier fact that a coarse equivalence (resp.ã coarse embedding) between the box spaces gives rise to a coarse equivalence (resp.ã coarse embedding) between the groups. We deduce new invariants that distinguish box spaces up to coarse embedding and coarse equivalence. In particular, we obtain that the expanders coming from [Formula: see text] cannot be coarsely embedded inside the expanders of [Formula: see text], where [Formula: see text] and [Formula: see text]. Moreover, we obtain a countable class of residually finite groups which are mutually coarse-equivalent but any of their box spaces are not coarse-equivalent.

Admitting a Coarse Embedding is Not Preserved Under Group Extensions

We construct a finitely generated group which is an extension of two finitely generated groups coarsely embeddable into Hilbert space but which itself does not coarsely embed into Hilbert space. Our construction also provides a new infinite monster group: the first example of a finitely generated group that does not coarsely embed into Hilbert space and yet does not contain a weakly embedded expander.

Random graphs, weak coarse embeddings, and higher index theory

This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse Baum–Connes assembly map is injective; the coarse Baum–Connes assembly map is not surjective; the maximal coarse Baum–Connes assembly map is an isomorphism. These results are closely tied to issues of expansion in graphs: in particular, we also show that such random sequences almost surely do not have geometric property (T), a strong form of expansion.The key geometric ingredients in the proof are due to Mendel and Naor: in our context, their results imply that a random sequence of graphs almost surely admits a weak form of coarse embedding into Hilbert space.