Related papers: Groups acting on hyperbolic $\Lambda$-metric space…

We prove a general criterion for a metric space to have conformal dimension one. The conditions are stated in terms of the existence of enough local cut points in the space. We then apply this criterion to the boundaries of hyperbolic…

Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Here we extend Ancona's potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schr\"odinger…

In this paper we introduce Patterson--Sullivan systems, which consist of a group action on a compact metrizable space and a quasi-invariant measure which behaves like a classical Patterson--Sullivan measure. For such systems we prove a…

We define the notion of a negatively curved tangent bundle of a metric measured space. We prove that, when a group $G$ acts on a metric measured space $X$ with a negatively curved tangent bundle, then $G$ acts on some $L^p$ space, and that…

In this paper we consider partial linear spaces induced on the point set of a polar space, but with as lines the hyperbolic lines of this polar space. We give some geometric characterizations of these and related spaces. The results have…

We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces, and strongly shortcut spaces. We show that every hierarchically hyperbolic space admits a new metric that is…

We develop the theory of Patterson-Sullivan measures on the boundary of a locally compact hyperbolic group, associating to certain left invariant metrics on the group measures on the boundary. We later prove that for second countable,…

We study several properties of expansive group actions on metric spaces and obtain relation between expansivity for subgroup and group actions. Through counter examples necessity of hypothesis are justified. We also study expansivity of…

We give a complete solution to the local classification program of higher rank partially hyperbolic algebraic actions. We show $C^\infty$ local rigidity of abelian ergodic algebraic actions for symmetric space examples, twisted symmetric…

We introduce a class of spaces, called real cubings, and study the stucture of groups acting nicely on these spaces. Just as cubings are a natural generalisation of simplicial trees, real cubings can be regarded as a natural generalisation…

Hyperbolic space and hyperbolic embeddings are becoming a popular research field for recommender systems. However, it is not clear under what circumstances the hyperbolic space should be considered. To fill this gap, This paper provides…

We investigate translation length functions for two-generated groups acting by isometries on $\Lambda$-trees, where $\Lambda$ is a totally ordered abelian group. In this context, we provide an explicit formula for the translation length of…

This paper investigates the notion of learning user and item representations in non-Euclidean space. Specifically, we study the connection between metric learning in hyperbolic space and collaborative filtering by exploring Mobius…

F. Paulin proved that if the Gromov boundaries of two hyperbolic groups are quasi-Mobius equivalent, then the groups themselves are quasi-isometric. The goal of this article is to extend Paulin's result to the setting of relatively…

The Alpha Group is an abstract geometry group in $\mathbb{R}^4$. The way it was conceived allows a new interpretation of the structure of hypercomplex space, with a new geometry and spatial topology, and a meaning for the geometric…

For any non-elementary hyperbolic group $\Gamma$, we find an outer automorphism invariant geodesic bicombing for the space of metric structures on $\Gamma$ equipped with a symmetrized version of the Thurston metric on Techim\"uller space.…

In 2006 Z. Sela and independently O. Kharlampovich and A. Myasnikov gave a solution to the Tarski problems by showing that two non-abelian free groups have the same elementary theory. Subsequently Z. Sela generalized the techniques used in…

This is an expository article on visual metrics on boundaries of hyperbolic metric spaces. We discuss the construction of visual metrics, quasisymmetries and their invariants, Hausdorff and conformal dimension, and constructions and…

The class of locally compact near abelian groups is introduced and investigated as a class of metabelian groups formalizing and applying the concept of scalar multiplication. The structure of locally compact near abelian groups and its…

In 1996, Gersten proved that finitely presented subgroups of a word hyperbolic group of integral cohomological dimension 2 are hyperbolic. We use isoperimetric inequalities over arbitrary rings to extend this result to any ring. In…