Related papers: Coarse median structures and homomorphisms from Ka…

We introduce the notion of strong embeddability for a metric space. This property lies between coarse embeddability and property A. A relative version of strong embeddability is developed in terms of a family of set maps on the metric…

This paper is devoted to introducing coarse structures in a very simple way, namely as an equivalence relation on the set of simple ends. As an application we show that Gromov boundary of every hyperbolic space is an example of a Higson…

We prove that quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid in the sense that they are uniformly close to product maps, this is a generalisation of the result obtained by Eskin, Fisher and Whyte in…

Cutting a hyperbolic surface X along a simple closed multi-geodesic results in a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to…

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the…

We give strengthened versions of the Herwig-Lascar and Hodkinson-Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous…

In this paper we develop a theory of Patterson--Sullivan measures associated to coarse cocycles of convergence groups. This framework includes Patterson-Sullivan measures associated to the Busemann cocycle on the geodesic boundary of a…

This paper studies coarse compactifications and their boundary. We introduce two alternative descriptions to Roe's original definition of coarse compactification. One approach uses bounded functions on $X$ that can be extended to the…

We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows…

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one…

We give a local characterization of the existence of Kazhdan projections for arbitary families of Banach space representations of a compactly generated locally compact group $G$. We also define and study a natural generalization of the Fell…

We study the geometry of the Thurston metric on the Teichm\"uller space $\mathcal{T}(S)$ of hyperbolic structures on a surface $S$. Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type;…

We prove asymptotically isometric, coarsely geodesic metrics on a toral relatively hyperbolic group are coarsely equal. The theorem applies to all lattices in SO(n,1). This partly verifies a conjecture by Margulis. In the case of hyperbolic…

We prove that the fundamental group of a finite graph of convergence groups with parabolic edge groups is a convergence group. Using this result, under some mild assumptions, we prove a combination theorem for a graph of convergence groups…

This paper aims to establish the geometrical finiteness for the natural isometric actions of (birational) automorphism groups on the hyperbolic spaces for K3 surfaces, Enriques surfaces, Coble surfaces, and irreducible symplectic varieties.…

The equivariant Gromov--Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by…

In this paper we classify countable locally finite-by-abelian groups up to coarse isomorphism. This classification is derived from a coarse classification of amenable shift-homogeneous metric spaces.

Let $\mathcal{X}_S$ denote the class of spaces homeomorphic to two closed orientable surfaces of genus greater than one identified to each other along an essential simple closed curve in each surface. Let $\mathcal{C}_S$ denote the set of…

We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal{K}$ of finite generalized metric spaces satisfies the Hrushovski extension property: for any…

We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group $G$ associated with non-singular $G$-spaces. We deduce that any two boundary representations of a…