Related papers: What is... hierarchical hyperbolicity?
Formal languages based on the multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups.
In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We…
There has been a great deal of research on graphs defined on algebraic structures in the last two decades. In this paper we begin an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this…
We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective)…
Continual learning has traditionally focused on classifying either instances or classes, but real-world applications, such as robotics and self-driving cars, require models to handle both simultaneously. To mirror real-life scenarios, we…
This is an introduction to the study of abstract homotopy theory by means of model categories and $(\infty,1)$-categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed.…
Looking to the fundamental domains of space groups we can investigate in which space they can be realized. If this space is hyperbolic, then the corresponding space group is also hyperbolic. In addition to the usual methods for…
We study the boundaries of relatively hyperbolic HHGs. Using the simplicial structure on the hierarchically hyperbolic boundary, we characterize both relative hyperbolicity and being thick of order 1 among HHGs. In the case of relatively…
In their first article, the authors initiated a systematic study of hyperbolic $\Lambda$-metric spaces, where $\Lambda$ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case $\Lambda =…
We give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are…
The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps.
This paper describes in basic terms what a "Thin Group" is, as well as its uses in various subjects.
In this paper, we introduce the concepts of short arc and length map in quasihyperbolic metric spaces, and obtain some geometric characterizations of Gromov hyperbolicity for quasihyperbolic metric spaces in terms of the properties of short…
In this paper, we generalize the well-known hyperbolic numbers to certain numeric structures scaled by the real numbers. Under our scaling of $\mathbb{R}$, the usual hyperbolic numbers are understood to be our 1-scaled hyperbolic numbers.…
Learning unbiased node representations for imbalanced samples in the graph has become a more remarkable and important topic. For the graph, a significant challenge is that the topological properties of the nodes (e.g., locations, roles) are…
Human proximity networks are temporal networks representing the close-range proximity among humans in a physical space. They have been extensively studied in the past 15 years as they are critical for understanding the spreading of diseases…
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 consider various equivalence relations on the set of homotopy classes of curves on a hyperbolic surface based on topological, algebraic, and geometric structures. The purpose of this work is to determine the relationship between these…
We classify groups G such that the unit group U(ZG) is hypercentral. In the second part we classify groups G whose modular group algebra has hyperbolic unit group V(KG).
The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.