- Research announcement: A combination theorem for acylindrical complexes of hyperbolic groups and Cannon-Thurston maps(arXiv)
Abstract : This is an announcement of some of the results obtained as a part of the second author’s Ph.D. thesis. In the first part, we prove that the fundamental group of an acylindrical complex of hyperbolic groups with finite edge groups is hyperbolic in which the vertex groups are quasiconvex. In the second part of the article, we prove the existence of Cannon-Thurston maps for certain subcomplexes of groups in acylindrical complexes of hyperbolic groups (see Theorem 0.4)
2. Cohomology fractals, Cannon-Thurston maps, and the geodesic flow(arXiv)
Abstract : Cohomology fractals are images naturally associated to cohomology classes in hyperbolic three-manifolds. We generate these images for cusped, incomplete, and closed hyperbolic three-manifolds in real-time by ray-tracing to a fixed visual radius. We discovered cohomology fractals while attempting to illustrate Cannon-Thurston maps without using vector graphics; we prove a correspondence between these two, when the cohomology class is dual to a fibration. This allows us to verify our implementations by comparing our images of cohomology fractals to existing pictures of Cannon-Thurston maps. In a sequence of experiments, we explore the limiting behaviour of cohomology fractals as the visual radius increases. Motivated by these experiments, we prove that the values of the cohomology fractals are normally distributed, but with diverging standard deviations. In fact, the cohomology fractals do not converge to a function in the limit. Instead, we show that the limit is a distribution on the sphere at infinity, only depending on the manifold and cohomology class
3.Pullbacks of metric bundles and Cannon-Thurston maps (arXiv)
Abstract : We introduce the notion of pullbacks of metric bundles. Given a metric (graph) bundle X over B where X and all the fibers are uniformly (Gromov) hyperbolic and nonelementary, and a Lipschitz qi embedding i:A→B we show that the pullback i∗X is hyperbolic and the map i∗:i∗X→X admits a continuous boundary extension, i.e. a Cannon-Thurston (CT) map ∂i∗:∂(i∗X)→∂X. As an application of our theorem we show that given a short exact sequence of nonelementary hyperbolic groups 1→N→G→πQ→1 and a finitely generated qi embedded subgroup Q1 4. Cannon-Thurston Maps(arXiv) Author : Mahan Mj Abstract : 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 groups and end with some open questions.
4. Cannon-Thurston Maps(arXiv)
Author : Mahan Mj
Abstract : 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 groups and end with some open questions.