Related papers: Conical limit sets of hyperbolic subgroups

We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using our methods we then prove that a finitely generated group $H$ admitting a quasi-isometric map $\phi$ into a…

In this article, we study acylindrical graphs of groups, local quasiconvexity, and Cannon-Thurston maps in the setting of totally disconnected locally compact (TDLC) hyperbolic groups, extending several fundamental notions and results from…

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result…

We explicate a number of notions of algebraic laminations existing in the literature, particularly in the context of an exact sequence $$1\to H\to G \to Q \to 1 $$ of hyperbolic groups. These laminations arise in different contexts:…

Given a metric (graph) bundle $X$ over $B$ where all the fibres are strongly relatively hyperbolic and nonelementary we show that, under certain conditions, $X$ is strongly hyperbolic relative to a collection of maximal cone-subbundles of…

Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition. Let $v$ be a vertex of $T$. Let $({X_v},d_v)$ denote the hyperbolic metric space corresponding to $v$. Then $i : X_v \rightarrow X$…

Given a group $G$ acting geometrically on a systolic complex $X$ and a hyperbolic isometry $h \in G$, we study the associated action of $h$ on the systolic boundary $\partial X$. We show that $h$ has a canonical pair of fixed points on the…

Let M be a compact hyperbolic manifold with totally geodesic boundary. If the injectivity radius of the boundary is larger than an explicit function of the normal injectivity radius of the boundary, we show that there is a negatively curved…

We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly.…

In this paper we provide a procedure to obtain a non-trivial HHS structure on a hyperbolic space. In particular, we prove that given a finite collection $\mathcal{F}$ of quasi-convex subgroups of a hyperbolic group $G$, there is an HHG…

In this paper we show that the existence of a non-parabolic local cut point in the Bowditch boundary $\partial(G,\mathbb{P})$ of a relatively hyperbolic group $(G,\mathbb{P})$ implies that $G$ splits over a $2$-ended subgroup. This theorem…

Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set,…

We give an overview of the theory of Cannon-Thurston maps which forms one of the links between the complex analytic and hyperbolic geometric study of Kleinian groups. We also briefly sketch connections to hyperbolic subgroups of hyperbolic…

We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain…

We prove the following boundary-theoretic characterization of relatively hyperbolic groups. Let $G$ be a finitely generated group with a finite collection $\mathcal{H}$ of finitely generated subgroups, and let $G^h$ denote the associated…

We prove that the canonical Thurston obstruction for a sub-hyperbolic semi-rational branched covering exists if the branched covering is not CLH-equivalent to a rational map.

We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in…

Baker and Riley proved that a free group of rank 3 can be contained in a hyperbolic group as a subgroup for which the Cannon-Thurston map is not well-defined. By using their result, we show that the phenomenon occurs for not only a free…

A group $\Gamma$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $\Gamma$ admits a cusp-uniform action on a proper $\delta$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric,…

Let $G$ be a group hyperbolic relative to a collection of subgroups $\{H_\lambda ,\lambda \in \Lambda \} $. We say that a subgroup $Q\le G$ is hyperbolically embedded into $G$, if $G$ is hyperbolic relative to $\{H_\lambda ,\lambda \in…