Related papers: Injectivity, cubical approximations and equivarian…

We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space. We apply this to characterize hyperbolically embedded subgroups…

Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. We introduce a topological notion of a sofic boundary attached to a given…

Local-to-global principles are spread all-around in mathematics. The classical Cartan-Hadamard Theorem from Riemannian geometry was generalized by W. Ballmann for metric spaces with non-positive curvature, and by S. Alexander and R. Bishop…

The paper presents a geometric duality between the spherical squared-Hellinger distance and a hyperbolic isometric invariant of the Poincare disc under the action of the general Mobius group. Motivated by the geometric connection, we…

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…

In this paper we establish new simple local geometric criteria for discrete entropic curvature introduced in [47] that are powerful enough to capture many geometric properties of complex models arising in mathematical physics. These results…

We develop the foundations of the theory of quasi-visual approximations of bounded metric spaces. Roughly speaking, these are sequences of covers of a given space for which the diameters of the sets in the covers shrink to zero and for…

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 use the projection complex machinery of Bestvina--Bromberg--Fujiwara to study hierarchically hyperbolic groups. In particular, we show that if the group has a BBF colouring and its associated hyperbolic spaces are quasiisometric to…

Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean…

This article is based on my lecture notes from summer schools at the Universities of Utah (June 2007) and Warwick (September 2007). We provide an introduction to explicit methods in the study of moduli spaces of quiver representations and…

In [arXiv:2008.04625] the authors constructed a classifying space for polystable holomorphic vector bundles on a compact K\"ahler manifold using analytic GIT theory. The aim of this article is to show that this classifying space taken in…

A general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact Hausdorff spaces. The procedure is then applied to gauge theories to carry out…

Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2-representations of finite groups on 2-Hilbert spaces, and their 2-characters. We show how the basic ideas of geometric quantization are…

In this paper we consider a large family of graphs of hierarchically hyperbolic groups (HHG) and show that their fundamental groups admit HHG structures. To do that, we will investigate the notion of hierarchical quasi convexity and show…

Representing data in hyperbolic space can effectively capture latent hierarchical relationships. With the goal of enabling accurate classification of points in hyperbolic space while respecting their hyperbolic geometry, we introduce…

We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension $d$ for all $d\geq 4$. These examples result from applying CAT$(0)$/CAT$(-1)$ filling constructions (based on singular doubly warped products) to…

For locally finite CAT(0) cube complexes it is known that they are injectively metrizable choosing the $l_\infty$-norm on each cube. In this paper we show that cube complexes which are injective with respect to this metric are always…

We introduce a new structure, the critical multi-cubic lattice. Notably the critical multi-cubic lattice is the first true generalization of the cubic lattice to higher dimensional spaces. We then introduce the notion of a homomorphism in…

Let $M$ be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group $\Gamma$ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover $\tilde M$ is a…