Related papers: Some groups with planar boundaries
We survey the known results regarding the boundaries of word-hyperbolic groups.
We study relatively hyperbolic group pairs whose boundaries are Schottky sets. We characterize the groups that have boundaries where the Schottky sets have incidence graphs with 1 or 2 components.
The aim of these notes is to connect the theory of hyperbolic and relatively hyperbolic groups to the theory of manifolds and Kleinian groups. We survey some of the extensive work that has been done in the field, and explain many examples.…
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…
We construct hyperbolic groups with the following properties: The boundary of the group has big dimension, it is separated by a Cantor set and the group does not split. This shows that Bowditch's theorem that characterizes splittings of…
We classify the boundaries of hyperbolic groups that have enough quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.
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 begin an investigation into the behavior of Bowditch and Gromov boundaries under the operation of Dehn filling. In particular we show many Dehn fillings of a toral relatively hyperbolic group with 2-sphere boundary are hyperbolic with…
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…
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…
We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an acylindrical amalgamation of relatively…
We review the theory of splittings of hyperbolic groups, as determined by the topology of the boundary. We give explicit examples of certain phenomena and then use this to describe limit sets of Kleinian groups up to homeomorphism.
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)…
In this paper, we prove a combination theorem for a relatively acylindrical graph of relatively hyperbolic groups (Theorem 1.1). Here, we are extending the technique of [Tom21] and constructing Bowditch boundary of the fundamental group of…
Carrier graphs of groups representing subgroups of a given relatively hyperbolic groups are introduced and a combination theorem for relatively quasi-convex subgroups is proven. Subsequently a theory of folds for such carrier graphs is…
In this paper we prove a general structure theorem for relatively hyperbolic groups (with arbitrary peripheral subgroups) acting naive convex co-compactly on properly convex domains in real projective space. We also establish a…
In this note we use Yaman's dynamical characterization of relative hyperbolicity to prove a theorem of Bowditch about relatively hyperbolic pairs $(G,\mathcal{H})$ with $G$ hyperbolic. Our proof additionally gives a description of the…
We exhibit examples of separable boundaries for non-hyperbolic groups. The main ingredient is the alignment property introduced by Furman in the study of rigidity properties of discrete subgroups of algebraic groups.
The topology of the Bowditch boundary of a relatively hyperbolic group pair gives information about relative splittings of the group. It is therefore interesting to ask if there is generic behavior of this boundary. The purpose of this…
We consider splittings of groups over finite and two-ended subgroups. We study the combinatorics of such splittings using generalisations of Whitehead graphs. In the case of hyperbolic groups, we relate this to the topology of the boundary.…